arXiv:2110.10647v1 [math.AP] 20 Oct 2021

LOW REGULARITY ILL-POSEDNESS AND SHOCK FORMATION

FOR 3D IDEAL COMPRESSIBLE MHD

XINLIANG AN, HAOYANG CHEN, AND SILU YIN

Abstract. The study of magnetohydrodynamics (MHD) significantly boosts the understand-ing and development of solar physics, planetary dynamics and controlled nuclear fusion. Dy-namical properties of the MHD system involve nonlinear interactions of waves with multipletravelling speeds (the fast and slow magnetosonic waves, the Alfven wave and the entropywave). One intriguing topic is the shock phenomena accompanied by the magnetic field, whichhave been affirmed by astronomical observations. However, permitting the residence of all abovemulti-speed waves, mathematically, whether one can prove shock formation for 3D MHD is stillopen. The multiple-speed nature of the MHD system makes it fascinating and challenging. Inthis paper, we give an affirmative answer to the above question. For 3D ideal compressible MHD,we construct examples of shock formation allowing the presence of all characteristic waves withmultiple wave speeds. Building on our construction, we further prove that the Cauchy problemfor 3D ideal MHD is H2 ill-posed. And this is caused by the instantaneous shock formation.In particular, when the magnetic field is absent, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to theregularity of the fluid velocity. Our proof for 3D MHD is based on a coalition of a carefullydesigned algebraic approach and a geometric approach. To trace the nonlinear interactions ofvarious waves, we algebraically decompose the 3D ideal MHD equations into a 7×7 non-strictlyhyperbolic system. Via detailed calculations, we reveal its hidden subtle structures. With themwe give a complete description of MHD dynamics up to the earliest singular event, when a shockforms.

Contents

1. Introduction 21.1. Main theorems 41.2. Difficulties and Strategies 101.3. Main steps in the proof 161.4. Other related works 191.5. Acknowledgements 192. Decomposition of waves 193. Proof of Theorem 1.1 244. Proof of shock formation (Theorem 1.2) 274.1. A priori estimates 274.2. L∞-estimates and bootstrap argument 344.3. Shock formation 365. Proof of H2 ill-posedness (Theorem 1.3) 395.1. Initial data designed for proving ill-posedness 395.2. Estimate for ∂z1ρ1 40

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5.3. Ill-posedness mechanism 436. Case of H1 = 0 457. Proof of H2 ill-posedness for 3D compressible Euler equations (Theorem 1.4) 47Appendix A. Proof of Proposition 3.1 49References 63

1. Introduction

Magnetohydrodynamics (MHD) plays a crucial role in astrophysics, planetary magnetismand controlled fusion reactor. It operates on every scales, from vast on stellar winds and solarmagnetic activity to the small on controlled fusion plasmas.

Mathematically, the equations of MHD system describe the dynamics and magnetic propertiesof electrically conducting fluid or plasma. To study planetary dynamics, liquid metals andelectrolyte, vibrant mathematical explorations of incompressible MHD equations are conducted,ranging from well-posedness theory to singularity formation mechanism and covering topics fromCauchy problems to free boundary problems.

Compared with the incompressible counterpart, there is less mathematical literature inves-tigating the compressible MHD. On the contrary, the physical phenomena governed by com-pressible MHD are extremely rich, which include the jets from accretion disks at a galacticscale, stellar magnetism describing solar activity, and controlled nuclear fusions in a designedtokamak.

Figure 1. Schematic of anaccretion disk. From [14] byDavidson.

Figure 2. The structure ofthe sun. Convection zone is thesource of MHD activity, includ-ing the initiation of coronal fluxtubes and sunspots. From [14] byDavidson.

Similarly to other compressible fluids with zero viscosity, shock waves are anticipated to play asignificant role in the context of 3D ideal compressible MHD with negligible resistivity/viscosity.For instance, a shock wave is expected to form when the solar wind interacts with the earth’smagnetic field.

ILL-POSEDNESS AND SHOCK FORMATION OF COMPRESSIBLE MHD 3

Figure 3. The interaction of the earth’s magnetic field with the solar wind.From [14] by Davidson.

Shock-wave-related phenomena of MHD raise broad interests among the physical and thenumerical communities. The evolution of compressible MHD involves interactions of waves withmultiple speeds: the fast and slow magnetosonic waves, the Alfven wave and the entropy wave.Mathematically, an open question is to give a rigorous proof of shock formation allowing theresidence of all above multi-speed waves.

In this paper, we provide a confirmative answer to the above question for the 3D ideal com-pressible MHD. We give a constructive proof including detailed analysis of the solutions’ be-haviours up to the first (shock) singularity. And the shock forms due to the compression ofcharacteristics. Furthermore, based on shock formation, we prove that the Cauchy problems ofthe 3D ideal compressible MHD equations are ill-posed in H2(R3). We show that the under-lying mechanism of this ill-posedness is the instantaneous shock formation. To the best of ourknowledge, this is also the first nonlinear result of low regularity ill-posedness for the 3D idealcompressible MHD. In the absence of the magnetic field, we also provide a desired ill-posednessresult for the 3D compressible Euler equations.

Our research toward low-regularity ill-posedness for MHD is inspired by remarkable workson wave equations. Under planar symmetry, Lindblad gave sharp counterexamples to the low-regularity local well-posedness for semilinear and certain quasilinear wave equations in [29–31].In [16], Ettinger-Lindblad generalized the above results to the Einstein’s equations and con-structed a sharp counterexample for local well-posedness of Einstein vacuum equations in wavecoordinates. An exploration by Granowski [17] later showed that Lindblad’s Hs ill-posednessin [31] is stable under general perturbations. Without any symmetry assumption, Alinhac [1,2]firstly proved shock formation for solutions to quasilinear wave equations in more than one spa-tial dimension. He employed a Nash-Moser iteration scheme to overcome the loss of derivativesin energy estimates. This approach relies heavily on a non-degeneracy condition posed on initialdata and fails to reveal information beyond the first blow-up point. In 2009, a breakthrough [10]was made by Christodoulou. Based on a fully geometric approach, he sharpened Alinhac’sresults by giving a detailed understanding and a complete description of shock formation for3D irrotational Euler equations. With the full use of the geometry therein, Chrsitodoulou de-rived sharper energy estimates which avoid the potential loss of derivatives. This geometricapproach was later extended to a larger class of wave equations and Euler equations with orwithout vorticity. See Christodoulou-Miao [11], Luk-Speck [35], Miao-Yu [37], Speck [39, 40],


Speck-Holzegel-Luk-Wong [41]. Recently, applying a different approach via self-similar ansatz,Buckmaster-Shkoller-Vicol [5, 6] constructed shock formation for compressible Euler equationsin 2D and 3D allowing the presence of non-trivial vorticity.

Compared with the single wave equation, fewer results are known for quasilinear wave systems.Quasilinear wave systems with physical background (e.g. elasticity and MHD) typically describephenomena involving interactions of waves with multiple travelling speeds. An important andchallenging task is to study singularity formation for these multi-wave-speed hyperbolic systems,including elastic waves, compressible MHD, crystal optics, self-gravitating relativistic fluids, etc.In [3], we generalized Lindblad’s work on a scalar wave equation in [31] by showing that theCauchy problems for 3D elastic waves, a physical system with multiple wave speeds, are ill-posed in H3(R3). We also quantitatively demonstrated that the ill-posedness is driven by shockformation. In this paper, we further generalize our method and result in [3] to an importantquasilinear hyperbolic system with multiple wave-speed: the 3D ideal compressible MHD system.Compared with the elastic waves, the MHD system possesses more subtle and more complexstructures, which imposes many new challenges to establish ill-posedness and shock formation.See Subsection 1.2 for details.

1.1. Main theorems. We start from inducing the equations. The 3D ideal compressible MHD,composed of the Euler equations of fluid dynamics and the coupled Maxwell’s equations formagnetic field, forms a quasilinear hyperbolic system:

∂t%+∇ · (%u) = 0,

%∂t + (u · ∇)u+∇p+ µ0H × rotH = 0,

∂tH − rot(u×H) = 0,

∂tS + (u · ∇)S = 0,

∇ ·H = 0.

(1.1)

Here, µ0 is the magnetic permeability constant, % is the fluid density1, u = (u1, u2, u3) ∈ R3

is the fluid velocity, H = (H1, H2, H3) ∈ R3 is the magnetic field intensity, S is the entropyand p is the pressure satisfying the equation of state p = p(%, S). To describe the solar wind,following [14] we use the polytropic equation of state

p = AeS%γ (1.2)

with A being a positive constant and γ > 1 being the adiabatic gas constant.Following Lindblad’s approach toward proving ill-posedness, we study (1.1) under planar

symmetry. By Gauss’s Law, H1 is always a constant. See Section 2. Therefore, we actuallywork with the following system:

∂tΦ(x, t) +A(Φ)∂xΦ(x, t) = 0, (1.3)

where Φ = (u1, u2, u3, % − 1, H2, H3, S)T and A(Φ) ∈ R7×7 is a 7 × 7 matrix. For notationalsimplicity, we denote x1 by x in this paper. Our proof relies on a carefully designed algebraicwave decomposition of this 7 × 7 system. See Subsection 1.3 and Section 2 for details. Thedelicate structures of the decomposed system play a crucial role in our proof. In the followingtheorem, these subtle structures are revealed for the first time.

1In our analysis, the value of % is close to 1.


Theorem 1.1 (Structures of MHD equations). For the MHD system (1.3): ∂tΦ(x, t) +A(Φ)∂xΦ(x, t) = 0, one can rewrite it into the following form:

∂siρi = ciiivi +( ∑m6=i

ciimwm

)ρi,

∂siwi = −ciiiw2i +

( ∑m 6=i

(−ciim + γiim)wm

)wi +

∑m 6=i,k 6=im 6=k

γikmwkwm,

∂sivi =( ∑m6=i

γiimwm

)vi +

∑m 6=i,k 6=im 6=k

γikmwkwmρi.

(1.4)

Here, ∂si = λi∂x+∂t with λi being the ith eigenvalue of the 7×7 matrix A(Φ). And ρi := ∂ziXi isthe inverse density of the characteristics2. We further define wi := li∂xΦ and vi := ρiwi, whereli is the ith left eigenvector of A(Φ). With an 18-page calculation, we prove that the coefficientsciii, c

iim, γ

iim, γ

ikm being non-trivial functions of Φ are all of order 1. (See Section 3 and Appendix

A for the details3.) This enables us to reveal the following structures for the system (1.4). Forρi, we have

∂s1ρ1 ∼v1 + ρ1(∑i 6=1

wi),

∂s2ρ2 ∼v2 + ρ2w3 + ρ2( ∑i 6=2,3

wi),

∂s3ρ3 ∼v3 + ρ3w2 + ρ3( ∑i6=2,3

wi),

∂s4ρ4 ∼v4 + ρ4(∑i6=4

wi),

∂s5ρ5 ∼v5 + ρ5w6 + ρ5( ∑i 6=5,6

wi),

∂s6ρ6 ∼v6 + ρ6w5 + ρ6( ∑i 6=5,6

wi),

∂s7ρ7 ∼v7 + ρ7(∑i 6=7

wi).

(1.5)

2We use (Xi(zi, si), si) to denote the trajectory of the characteristics with (zi, si) being the characteristiccoordinates. See (2.12) for detailed definitions.

3These coefficients could potentially be singular due to the singular factors present in the eigenvectors.


Here, the notation v2 denotes the absence of the term v2, ρ2w3 denotes the strong interaction term.

For wi, we have

∂s1w1 ∼(w1)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +∑

m6=1,k 6=1,m 6=k(m,k)6=(2,3),(m,k) 6=(5,6)

wmwk,

∂s2w2 ∼(w2)2 + w2w3 + (λ5 − λ6)w5w6 +∑

m6=2,k 6=2,m 6=k(m,k)6=(5,6)

wmwk,

∂s3w3 ∼(w3)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +∑

m6=3,k 6=3,m 6=k(m,k) 6=(5,6)

wmwk,

∂s4w4 ∼(w4)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m6=4,k 6=4,m 6=k

(m,k) 6=(2,3),(m,k)6=(5,6)

wmwk,

∂s5w5 ∼(w5)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +∑

m6=5,k 6=5,m 6=k(m,k) 6=(5,6)

wmwk,

∂s6w6 ∼(w6)2 + (λ2 − λ3)w2w3 + w5w6 +∑

m6=6,k 6=6,m 6=k(m,k)6=(2,3)

wmwk,

∂s7w7 ∼(w7)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +∑

m6=7,k 6=7,m 6=k(m,k)6=(2,3),(m,k) 6=(5,6)

wmwk.

(1.6)

Here, the notation (λ2 − λ3)w2w3 denotes the weak interaction term. For vi, we have

∂s1v1 ∼v1(∑i 6=1

wi)

+ ρ1

((λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m6=1,k 6=1,m 6=k

(m,k)6=(2,3),(m,k)6=(5,6)

wmwk

),

∂s2v2 ∼ (λ2 − λ3)v2w3 +( ∑i6=2,3

wi)v2 + ρ2

((λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m 6=2,k 6=2,m 6=k(m,k)6=(5,6)

wmwk

),

∂s3v3 ∼ (λ2 − λ3)v3w2 +( ∑i6=2,3

wi)v3 + ρ3

((λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m 6=3,k 6=3,m 6=k(m,k)6=(5,6)

wmwk

),

∂s4v4 ∼v4(∑i 6=4

wi)

+ ρ4

((λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m6=4,k 6=4,m 6=k

(m,k)6=(2,3),(m,k)6=(5,6)

wmwk

),

∂s5v5 ∼ (λ5 − λ6)v5w6 +( ∑i6=5,6

wi)v5 + ρ5

((λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m 6=5,k 6=5,m 6=k(m,k)6=(2,3)

wmwk

),

∂s6v6 ∼ (λ5 − λ6)v6w5 +( ∑i6=5,6

wi)v6 + ρ6

((λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m 6=6,k 6=6,m 6=k(m,k)6=(2,3)

wmwk

),

∂s7v7 ∼v7(∑i 6=7

wi)

+ ρ7

((λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 +

∑m6=7,k 6=7,m 6=k

(m,k)6=(2,3),(m,k)6=(5,6)

wmwk

).

(1.7)


Remark 1.1 (Potentially harmful terms). In (1.6)-(1.7), some potentially harmful termsare present. We list them on the right hand side of the following equations

∂s2w2 = (γ223 − c2

23)w2w3 + · · · , (1.8)

∂s6w6 = (γ665 − c6

65)w5w6 + · · · , (1.9)

∂s2ρ2 = c223ρ2w3 + · · · , (1.10)

∂s6ρ6 = c665ρ6w5 + · · · . (1.11)

These potentially harmful terms could result in strong nonlinear interactions between the twofamilies of waves (w2 and w3, w5 and w6) with almost-the-same speeds. To handle these nonlin-ear terms is the main task of Section 4. Compared with elastic waves in [3], there the coefficientsof these potentially harmful nonlinear terms have additional smallness, which makes the inter-actions weaker (and thus easier to control) than those in MHD.

Remark 1.2. In Theorem 1.1, our conclusion that all ciii(Φ), ciim(Φ), γiim(Φ), γikm(Φ) are oforder 1 is highly non-trivial. This is true because we carefully design the wave decomposition inthis paper and all the potentially singular factors in ciii(Φ), ciim(Φ), γiim(Φ), γikm(Φ) are cancelled.To prove this, we conduct an 18-page (mainly in the appendix) calculation for 299 coefficients.See Proposition 3.1 in Section 3 and Appendix A for the crucial discussions.

We are now in a good position to state the following result: we construct examples of shockformation for 3D ideal compressible MHD equations allowing the presence of nonlinear interac-tions of all multi-speed waves.

Theorem 1.2 (Shock formation for MHD). For the 3D ideal compressible MHD, thereexists a large class of compactly supported smooth Cauchy initial data that lead to finite-timeshock formations. Let ε ∈ (0, 1

100) be a fixed parameter. Then there is a constant θ0 > 0 suchthat for any 0 < θ < θ0, the following statements hold for the solution to system (1.4) arisingfrom the initial datum.

i) (Shock formation along characteristics) A shock forms at time T ∗, i.e.,

limt→T ∗

ρ1(z0, t) = 0 and limt→T ∗

w1(z0, t) =∞,

where at z0 the initial datum w1(z1, 0) attains its maximum. In particular, the blow-upof w1 implies limt→T ∗ |∂xΦ|(z0, t) =∞. The lifespan T ∗ of w1 satisfies

1

(1 + ε)3|c111(0)|w1(z0, 0)

≤ T ∗ ≤ 1

(1− ε)3|c111(0)|w1(z0, 0)

. (1.12)

ii) (Lower bound of other ρi) For any i ∈ 2, · · · , 7, we have wi(zi, t) = O(θ) andρi(zi, t) > C with C being a uniform positive constant for all zi ∈ R and any t < T ∗.

iii) (Boundedness of Φ) The original variable Φ is uniformly bounded. We have Φ(zi, t) =O(θ) for all zi ∈ R and any t < T ∗.

iv) (Boundedness of ρi and vi) For any i ∈ 1, · · · , 7, we have ρi(zi, t) = O(1) andvi(zi, t) = O(θ) for all zi ∈ R and any t < T ∗.


Remark 1.3. For the decomposed system (1.4), initial data are compactly supported on [−2η, 2η]with η > 0. Then, along different families of characteristics, there are seven strips Ri, i = 1 · · · 7issuing from the support [−2η, 2η]. To conduct our analysis, we use the following picture. Theshock is constructed to happen at t = T ∗η in the first characteristic strip R1 with the fastestpropagation wave speed. For t < T ∗η , there is no other singular point in the spacetime region andthe solution is smooth. See Figure 4.

2η−2η t = 0

t = t(η)0

t = T ∗η

z0

(z0, T∗η )

R1R2 ∪R3R4R5 ∪R6R7

Figure 4. First shock forms at (z0, T∗η ) in the characteristic strip R1. In this

picture, the domain of our consideration is divided into three regions: (a) The grey regionis called the characteristic strips. (b) The light grey region denotes the disjoint domainbetween two separate strips. (c) The dark grey region denotes the domain where thestrips overlap with each other.

Remark 1.4. We design the above strips R1,R2∪R3,R4,R5∪R6,R7 and regions with colorsto conduct our analysis. We remark that the solution is not necessarily being zero outside of thecharacteristic strips. In particular, the solutions’ dynamics in the light-grey region of Figure 4is far from being trivial. For our proof, it is crucial to investigate the different behaviours of thesolutions in and out of these characteristic strips. We will derive a priori estimates for solutionsin these colored regions in Subsection 4.1.

Moreover, based on our constructed shock formation, we further provide a counterexample toH2 local well-posedness. It is caused by the instantaneous shock formation. The ill-posednesstheorem is stated as below.

Theorem 1.3 (H2 ill-posedness for MHD). The Cauchy problems of the 3D ideal compress-ible MHD equations (1.1) are ill-posed in H2(R3) in the following sense:

There exists a family of compactly supported, smooth initial data Φ(η)0 satisfying

‖Φ(η)0 ‖H2(R3) → 0 as η → 0,

where η > 0 is a small parameter and it identifies the datum in this family. For each η, theCauchy problem of the 3D ideal MHD (1.3) admits a solution that ceases to be regular in finitetime. Let T ∗η be the largest time such that the solution Φη ∈ C∞

(R3 × [0, T ∗η )

). The following

statements hold:

i) (Instantaneous shock formation) Evolving from each initial datum Φ(η)0 , a shock

forms at T ∗η . And as η → 0, we have T ∗η → 0. This means that the constructed initial

data Φ(η)0 give rise to an instantaneous shock formation.


ii) (Blow-up of H1-norm) For each solution Φη, with ΩT ∗η being a spatial neighborhood

of the first (shock) singularity. the H1-norm ‖Φη‖H1(ΩT∗η ) blows up at T ∗η . In particular,

‖∂xu(η)1 (·, T ∗η )‖L2(ΩT∗η ) = +∞, ‖∂x%(η)(·, T ∗η )‖L2(ΩT∗η ) = +∞. (1.13)

Remark 1.5. Noted that the L∞ blow-up of the solutions’ first derivatives in Theorem 1.2 is notadequate to imply the blow-up of the H1 norm as in (1.13). To prove (1.13), we crucially employthe vanishing of the inverse density ρ, which is the geometric description of the shock formation.Our exploration of the characteristics’ geometric behaviours via ρ bridges the gap between theshock formation (pointwise blow-up of the first derivatives) and the blow-up of H1-norms. SeeSubsection 5.3 for the details.

Remark 1.6. For 3D ideal compressible MHD (1.1), both our examples of the shock formationallowing the presence of waves with multiple travelling speeds and the associated low-regularityill-posedness are new. With the subtle structures we found, for our constructed solutions, we givea complete description of the MHD dynamics up to the time T ∗η when the first (shock) singularityhappens.

If the magnetic field vanishes, i.e., H1 = H2 = H3 ≡ 0, our system (1.1) reduces to the 3Dcompressible Euler equations. Our results thus give

Theorem 1.4 (H2 ill-posedness for Euler equations). Consider the following 3D compress-ible Euler equations allowing non-trivial entropy and vorticity

∂t%+∇ · (%u) = 0,

%∂t + (u · ∇)u+∇p = 0,

∂tS + (u · ∇)S = 0.

(1.14)

Then the Cauchy problem of (1.14) is H2 ill-posed with respect to the fluid velocity and den-sity. More precisely, there exits a family of compactly supported smooth initial data for (1.14)satisfying

‖%(η)0 ‖H2(R3) + ‖u(η)

0 ‖H2(R3) + ‖S(η)0 ‖H2(R3) → 0 as η → 0.

After a finite time T ∗η a shock forms and the solution ceases to be smooth. Furthermore, tied tothis family of initial data the shock formation is instantaneous, i.e., T ∗η → 0 as η → 0. Moreover,

the H1-norm of the velocity u1 and density % blow up at the shock formation time T ∗η :

‖u(η)1 (·, T ∗η )‖H1(ΩT∗η ) = +∞, ‖%(η)(·, T ∗η )‖H1(ΩT∗η ) = +∞

where ΩT ∗η is a spatial region around the first (shock) singularity.

Remark 1.7. The above ill-posedness result for the 3D compressible Euler equations is sharpwith respect to the regularity of the fluid velocity u and density %. The classical local well-posedness (LWP) result for the 3D compressible Euler equations holds in Hs with s > 5

2 forinitial data of u and %. The regularity for local well-posedness is lowered to s > 2 for fluid velocityand density by Disconzi-Luo-Mazzone-Speck [13], Wang [46] and Zhang-Andersson [52]. Ourill-posedness result thus acts as a sharp counterpart of these local well-posedness results. As aspecial scenario of 3D compressible Euler equations, for the incompressible case, Bourgain-Li [4]proved the sharp low-regularity ill-posedness result with respect to the regularity of the vorticity


ω. Their initial data require fluid velocity u and vorticity ω satisfying (u, ω) ∈ H52 ×H

32 . With

the picture below, one can see that our Theorem 1.4 serves as a sharp companion with respectto the fluid velocity u and density %. See Section 7 for more detailed discussions.

s: regularity for the velocity

s′: regularity for the vorticity

2 52

32

2

LWP proved by Wang [46]

(See also Disconzi-Luo-Mazzone

-Speck [13], Zhang-Andersson [52])

LWP by the Conjecture in [46]

Bourgain-Li’s ill-posedness [4]

An-Chen-Yin’s

ill-posedness

1.2. Difficulties and Strategies. To prove the ill-posedness result, as in [3,16,30,31] we workunder planar symmetry, the MHD system then obeys

%∂tu1 + %u1∂xu1 + c2∂x%+ ∂Sp∂xS + µ0H2∂xH2 + µ0H3∂xH3 = 0,

%∂tu2 + %u1∂xu2 − µ0H1∂xH2 = 0,

%∂tu3 + %u1∂xu3 − µ0H1∂xH3 = 0,

∂t%+ u1∂x%+ %∂xu1 = 0,

∂tH2 + u1∂xH2 −H1∂xu2 +H2∂xu1 = 0,

∂tH3 + u1∂xH3 −H1∂xu3 +H3∂xu1 = 0,

∂tS + u1∂xS = 0,

(1.15)

where c =√∂%p denotes the sound speed. For (1.15), by Gauss’s Law, H1 is a constant and

hence (1.15) can be viewed as a 7× 7 first-order hyperbolic system:

∂tΦ +A(Φ)∂xΦ = 0, (1.16)

where Φ = (u1, u2, u3, % − 1, H2, H3, S)T and A(Φ) is a 7 × 7 matrix. We then explore thestructures of (1.16) and proceed to do analysis, but several difficulties arise:• Invalidity of Riemann invariants for a 7 × 7 system: For hyperbolic system with

different travelling speeds, to prove shock formation, a key ingredient is to understand thenonlinear interaction of waves propagating in different speeds. For a 2× 2 first-order hyperbolicsystem, there exist two Riemann invariants, which can be constructed explicitly and verifytwo transport equations. These two Riemann invariants carry crucial information about thenonlinear structure of the 2 × 2 first-order hyperbolic system. And with them other geometricquantities can be solved along the characteristics. However, for a large system, such as our 7×7system, it is almost impossible to find proper Riemann invariants with explicit formulas. Herewe adopt a different approach. We appeal to John’s classic method [25], i.e., decomposition ofwaves. This approach was used by us in [3] to study the elastic waves equations, and it will


further be extended to the 7× 7 MHD system in this paper. Applying the following formula ofdecomposition of waves

∂xΦ =7∑

k=1

wkrk,

we rewrite (1.16) as a diagonal Riccati-type system:

(∂t + λi∂x)wi = −ciii(wi)2 +(∑m 6=i

(−ciim + γiim)wm

)wi +

∑m 6=i,k 6=im 6=k

γikmwkwm, (1.17)

where wi := li(Φ)∂xΦ and li(Φ) are the left eigenvectors of A(Φ). Here the coefficients ciii, ciim,

γikm, γiim are functions of the unknowns Φ. A fundamental technical point in our work is tobound these coefficients. This is closely relevant to the choice of right eigenvectors ri.• Non-strict hyperbolicity: For the hyperbolic system (1.16), via algebraic calculations, we

notice that the MHD system (1.15) is non-strictly hyperbolic4, with eigenvalues of the coefficientmatrix (A(Φ))7×7 satisfying

λ7(Φ) < λ6(Φ) ≤ λ5(Φ) < λ4(Φ) < λ3(Φ) ≤ λ2(Φ) < λ1(Φ).

When Φ is sufficiently small, two pairs of characteristics have almost-the-same travelling speedλ2(Φ) ≈ λ3(Φ), λ5(Φ) ≈ λ6(Φ). This yields the long-time overlapping of the corresponding char-acteristic strips R2,R3 (or R5,R6) mentioned in Remark 1.3. Substantial nonlinear interactionswill be carried out in the overlapped regions and they may effect other strips. We overcome thisobstacle by taking unions of the overlapped strips and considering the wave propagations in andout of the following five strips:

R1,R2 ∪R3,R4,R5 ∪R6,R7. (1.18)

Tracing the dynamics of MHD (a 7 × 7 non-strictly hyperbolic system) with analyzing thesolutions’ behaviours in and out of these five characteristic strips R1,R2∪R3,R4,R5∪R6,R7is new. These five strips will completely separate from each other when t > t

(η)0 , where t

(η)0 can

be precisely calculated. See the following picture sketching the dynamics:

2η−2η t = 0

t = t(η)0

R1R2 ∪R3R4R5 ∪R6R7

Figure 5. Separation of five characteristic strips.

4For more discussions on non-strictly hyperbolic system, we refer to Liu [33], Liu-Xin [34] and Zhou [54].


Estimates before t(η)0 can be carried out similarly as in [12] by Christodoulou-Perez, where

they studied the propagation of electromagnetic waves in nonlinear crystals and solved a first-

order strictly hyperbolic system. However, after t(η)0 , the non-strict hyperbolicity dramatically

amplifies the nonlinear interactions. R2 and R3 (or R5 and R6) could overlap with each otherfor a long time. This overlapping feature results in strong nonlinear interaction. Here weillustrate the intuition with the interactions in R2 and R3. In strictly hyperbolic case as in [12],

since R2 and R3 will separate after t(η)0 , the interaction of corresponding characteristic waves

is weak. However, for our non-strictly hyperbolic system, due to the overlapping feature, thecharacteristics C2, C3 may still intersect in R2 (or R3) after a long time. This forces us toestimate in R2 ∪ R3 where the interaction is conceivably stronger. Nonetheless, we show thatthe strong interaction will not jeopardize the desired blow-up mechanism. To guarantee that theinteraction terms are still controllable, we need to employ the subtle structures of the equations.This will be discussed in the following paragraph.• Subtle structures in MHD system have to be employed: The subtleties lying in

the structures of (1.17) are of paramount importance in our proof. A thorough analysis of thestructures is essential for tracing the geometric behaviours of the characteristics in and out ofthe above five strips (1.18) up to shock formation time T ∗η . We list some crucial structures forwi here:

(∂t + λ1∂x)w1 ∼(w1)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 + · · · ,

(∂t + λ2∂x)w2 ∼(w2)2 + w2w3 + (λ5 − λ6)w5w6 + · · · ,

(∂t + λ3∂x)w3 ∼(w3)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 + · · · ,

(∂t + λ4∂x)w4 ∼(w4)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 + · · · ,(∂t + λ5∂x)w5 ∼(w5)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 + · · · ,

(∂t + λ6∂x)w6 ∼(w6)2 + (λ2 − λ3)w2w3 + w5w6 + · · · ,

(∂t + λ7∂x)w7 ∼(w7)2 + (λ2 − λ3)w2w3 + (λ5 − λ6)w5w6 + · · · .

(1.19)

The coefficients of the deleted terms in (1.19) are proved to be zero. In other words, there is noRiccati-type terms appearing in the equations of wii=2,4,6. The vanishing of these terms indi-cates the linear degeneracy of the corresponding characteristics. In contrast, the non-vanishingof Riccati-type terms such as (w1)2, (w3)2, (w5)2, (w7)2 demonstrate the genuine nonlinearity ofthe corresponding characteristics and they may drive the blow-ups of the solutions.

Now we compare with the circumstance for the elastic waves in [3]. The system of wi in [3]reads

(∂t + λ1∂x)w1 ∼(w1)2 + w2w3 + w4w5 + · · · ,

(∂t + λ2∂x)w2 ∼(w2)2 + (λ2 − λ3)w2w3 + (λ4 − λ5)w4w5 + · · · ,(∂t + λ3∂x)w3 ∼(w3)2 + w2w3 + w4w5 + · · · ,(∂t + λ4∂x)w4 ∼(w4)2 + w2w3 + w4w5 + · · · ,

(∂t + λ5∂x)w5 ∼(w5)2 + (λ2 − λ3)w2w3 + (λ4 − λ5)w4w5 + · · · ,(∂t + λ6∂x)w6 ∼(w6)2 + w2w3 + w4w5 + · · ·

(1.20)

where λ2 ≈ λ3 and λ5 ≈ λ6.


In the equations for w2 and w5 in (1.20), observe that the coefficients of w2w3 and w4w5

contain small factors λ2−λ3 and λ4−λ5 owing to the non-strict hyperbolicity. This suggests thatthe corresponding related interactions are weak. We were able to employ the bi-characteristiccoordinates transformation (x, t) =

(Xj(yj , t

′(yi, yj)), t′(yi, yj)

)to control these terms. For the

equation of w2, we integrate along C2 inside R3 to deduce

∫C2

∣∣(λ2 − λ3)w2

(X2(z2, t

′), t′)w3

(X2(z2, t

′), t′)∣∣dt′

≤∫y3∈[η,2η]

∣∣∣w2w3

(y3, t

′(y2, y3))(λ2 − λ3)

ρ3

(y3, t

′(y2, y3))

λ2 − λ3

∣∣∣dy3.

The only singular factor in the above integral is 1λ2−λ3

, which comes from the coordinate trans-formation. As we can see, it can be exactly cancelled by the small factor λ2−λ3 in the coefficient.

However, for the ideal compressible MHD, equations of w2 and w6 in (1.19) do not includesmall factor λ2−λ3 or λ5−λ6 in front of w2w3 or w5w6, which renders our approach in [3] via thebi-characteristic coordinates to fail. A more delicate choice of the initial data for w2, w3, w5, w6

and additional a priori estimates for

W (t) := max

sup(x′,t′)∈R2∪R3,

0≤t′≤t

w2, w3, sup(x′,t′)∈R5∪R6,

0≤t′≤t

w5, w6.

are employed to handle these terms.Moreover, for (1.19), the interaction terms w2w3 and w5w6 also appear in the equations of

w1, w3, w4, w5, w7. While the corresponding terms vanish in (1.20) since their coefficients areall zero. Luckily, in (1.19) we can prove these nonlinear interactions are weak and negligiblecompared with the ones in the equations of w2 and w6. In particular, the interaction termsw2w3, w5w6 are proved to be small compared with the dominant Riccati terms since they havesmall coefficients λ2 − λ3, λ5 − λ6. And the smallness here enables us to use bi-characteristiccoordinates to control w2w3, w5w6 with desired bounds.• Delicate choice of right eigenvectors: To explore the subtle structures of our system,

we need to control the following non-trivial coefficients ciim, γiim and γikm in the decomposedsystem (1.4), where

ciim := ∇Φλi · rm,γiim := −(λi − λm)li · (∇Φri · rm −∇Φrm · ri), m 6= i,

γikm := −(λk − λm)li · (∇Φrk · rm), k 6= i, m 6= i.

Here, Φ = (u1, u2, u3, %− 1, H2, H3, S)T . And λi, li, ri are the eigenvalues, left and right eigen-vectors of A(Φ), respectively. If all these coefficients are of O(1), the system will then take theschematic form presented in (1.19) and in Theorem 1.1. As we mentioned before, the realizationof these estimates for ciim, γiim, γikm relies heavily on carefully designed right eigenvectors ri. For


example, to evaluate γ264, one attempt is to take the following right eigenvectors

r2 =

0CaH3H1H2

−CaH1

0

−H3H2

10

, r4 =

000− %γ

001

, r6 =

0CaH1

−CaH2H1H3

01

−H2H3

0

, (1.21)

where Ca is the Alfven wave speed (See Section 2), and the left eigenvector

l2 = (0,H1H2H3

2Ca(H22 +H2

3 ),− H2

2H1

2Ca(H22 +H2

3 ), 0,− H2H3

2(H22 +H2

3 ),

H22

2(H22 +H2

3 ), 0). (1.22)

Here H1 6= 0 and the elements in l2 are of order 1. And it follows

γ264 = −(λ6 − λ4)l2 · (∇Φr6 · r4) = −(λ6 − λ4)H2

4γH3. (1.23)

However, since both H2 and H3 are unknowns with values close to 0, the term H2H3

is veryproblematic and may exhibit unexpected singularity. It may render the failure of the wholeargument. To overcome this problem, we renormalize and design the right eigenvectors and lefteigenvectors in the following form

r2 =

0CaH3H1

−CaH2H1

0−H3

H2

0

, r4 =

000− %γ

001

, r6 =

0CaH3H1

−CaH2H1

0H3

−H2

0

,

l2 = (0,H1H3

2Ca(H22 +H2

3 ),− H2H1

2Ca(H22 +H2

3 ), 0,− H3

2(H22 +H2

3 ),

H2

2(H22 +H2

3 ), 0).

Note that the right eigenvectors are more regular, but the left eigenvectors are more singular,since they always satisfy lirj = δij and lirj = δij . Miraculously, with the new set of r2, r4, r6

and l2, we have

γ264 = −(λ6 − λ4)l2 · (∇Φr6 · r4)

= −(λ6 − λ4)l2 ·(

0,CaH3

2H1γ,−CaH2

2H1γ, 0, 0, 0, 0

)>= −(λ6 − λ4)

( H23

4γ(H22 +H2

3 )+

H22

4γ(H22 +H2

3 )

)= −λ6 − λ4

4γ= O(1).


Compared with (1.23), the singular term is cancelled! Note that in l2, a potentially singularfactor 1

H22+H2

3appears since H2 and H3 are close to zero. But with a detailed matrix multipli-

cation, the singular denominator is cancelled! Not only for γ264, we have checked 299 terms with

18 pages calculations and find that with our designed right and left eigenvectors in (2.9) and(2.11), all these coefficients ciim, γiim and γikm are uniformly bounded.• Control of nonlinear interactions of multiple waves: Due to the multiple-wave-speed

trait of the MHD, the decomposed system (1.4) exhibits nonlinear interactions among differentcharacteristic strips. The high complexity of the coupled nonlinear terms poses a major technical

difficulty, especially after the separating time t(η)0 . To disclose how different waves interact with

each other, we must distinguish its influence to other characteristic waves in its own strips andin the exterior. Firstly, outside of the strips, we estimate the following quantities

Vi(t) := sup(x′,t′)/∈Ri,

0≤t′≤t

|wi(x′, t′)|, for i = 1, 4, 7,

V2(t) := max sup(x′,t′)/∈R2∪R3,

0≤t′≤t

|w2(x′, t′)|, |w3(x′, t′)|,

V5(t) := max sup(x′,t′)/∈R5∪R6,

0≤t′≤t

|w5(x′, t′)|, |w6(x′, t′)|,

V (t) := maxiVi(t) for i = 1, 2, 4, 5, 7.

Note that V2 and V5 are defined in a form different from that of V1, V4, V7. This is because, aswe emphasized before, for our non-strictly hyperbolic system, the almost-repeated characteristicwaves w2, w3 (or w5, w6) are treated together in R2∪R3 (or R5∪R6). We then prescribe initialdata such that only w1 will blow up. This is feasible since the equation of w1 is of Riccati type.Yet it is still hard to solve the Riccati equation directly in the presence of other nonlinear terms.To overcome this difficulty, we introduce the geometric quantities—the inverse densities ρi to

characterize the compression of the ith characteristics. To control wi, we appeal to investigatingdynamics of ρi and vi := ρiwi instead by estimating

Si(t) := sup(z′i,s′i)

z′i∈[−2η,2η], 0≤s′

i≤t

ρi(z′i, s′i), Ji(t) := sup

(z′i,s′i)

z′i∈[−2η,2η] 0≤s′

i≤t

|vi(z′i, s′i)|.

For the purpose of demonstration, we use ρ1 as an example. To bound ρ1, we estimate S1 viaintegrating (2.23)

∂ρ1

∂s1= O(J1 + V S1).

The first step then is to estimate J1. We integrate (2.25)

∂v1

∂s1= O(V J1 + V 2S1).

Here, it is important to analyze the behaviours of the nonlinear terms on the right. The nextstep is to investigate Vi. Since a priori we do not have the estimates for the solution’s life spanT , we employ the bi-characteristic coordinates to control Vi. Note that to control ρ1 in R1,we carefully estimate J1 and all Vi simultaneously. Compared with controlling ρ1 in R1, the


situation in the overlapped strips is more intricate. We refer the readers to Section 4.1 for moredetails.• Meaning of ill-posedness: For ideal compressible MHD (1.1), we further prove the H2

ill-posedness in 3D and we also demonstrate that the ill-posedness is driven by shock formation.In particular, the H2-norm of the solutions in 3D blows up at the shock formation time T ∗η withT ∗η → 0 as η → 0.

Based on our decomposition of waves, we trace the evolution of inverse densities ρi (i =1, · · · , 7). In R1, we prove that ρ1(z0, t) → 0 as t → T ∗η which renders T ∗η to be the first shockformation time. And ρ1 stays positive outsideR1. Then, with a uniform bound of ∂z1ρ1 obtainedin Subsection 5.2, in a constructed spatial region ΩT ∗η , we prove that

‖Φη(·, T ∗η )‖2H1(ΩT∗η ) ≥ Cη∫ z∗0

z0

1

ρ1(z, T ∗η )dz

(∗)= Cη

∫ z∗0

z0

1

ρ1(z, T ∗η )− ρ1(z0, T ∗η )dz

≥ Cη∫ z∗0

z0

1

(z − z0) sup |∂zρ1|dz

≥ Cη∫ z∗0

z0

1

z − z0dz = +∞.

As we mentioned in Remark 1.5, one can see from equality (∗) that the vanishing of ρ1 cruciallydrives the above blow-up. Moreover, for our choice of initial data, we have T ∗η → 0 as η → 0.These conclusions reveal a hidden mechanism: the instantaneous shock formation underlies theH2 ill-posedness.

1.3. Main steps in the proof. We outline our proof of Theorem 1.2-1.1 in this subsection.Step 1: Decomposition of waves (Section 2). Our first step is to algebraically decompose

the non-strictly hyperbolic 7 × 7 system (1.16). In this paper, we will use both characteristiccoordinates (zi, si) and bi-characteristic coordinates (yi, yj). Let Ci(zi) be the ith characteristic

with propagation speed λi starting at zi. Denote the corresponding ith characteristic strip to beRi := ∪zi∈I0Ci(zi), where I0 is the support of initial data. Define the inverse density of the ith

characteristics

ρi := ∂ziXi.

For fixed i, let

wi := li∂xΦ, and vi := ρiwi.

Then, with the left and right eigenvectors satisfying lirj = δij , one can decompose the originalsystem in the following way

∂xΦ =∑k

wkrk.

One can verify that these quantities satisfy

∂siρi =ciiivi +(∑m6=i

ciimwm

)ρi, (1.24)

∂siwi =− ciii(wi)2 +(∑m 6=i

(−ciim + γiim)wm

)wi +

∑m 6=i,k 6=im 6=k

γikmwkwm, (1.25)


∂sivi =(∑m 6=i

γiimwm

)vi +

∑m 6=i,k 6=im 6=k

γikmwkwmρi. (1.26)

Step 2: Analysis of structures (Section 3). We explore the subtle structures of the abovedecomposed system (1.24)-(1.26) by carrying out a thorough calculation of all the coefficientsciii, c

iim, γ

iim, γ

ikm. Firstly, recall that ciii being non-vanishing corresponds to the genuine non-

linearity. We show that c111 < 0 in Lemma 3.1. Hence the genuinely nonlinear condition is

verified and w1 satisfies a Riccati-type equation. In fact, the blow-up of w1 builds on the genuinenon-linearity. Then, we provide an explicit detailed computation of all coefficients. Based onour calculation, the decomposed system admits a schematic structure (1.5)-(1.7). We presentthe key ideas in Section 3. The Appendix A, a 15-page concrete calculation of the system’sstructure, supplies all ingredients to complete the proof of Theorem 1.1.

Step 3: A priori estimates (Subsection 4.1). For the decomposed system (1.4), charac-teristic waves in different Ri interact with each other. In particular, two pairs of characteristicstrips (R2 and R3, R5 and R6) may overlap for a long time which yields strong nonlinearinteractions. Our strategy is to study the dynamics of waves in the following five strips:

R1,R2 ∪R3,R4,R5 ∪R6,R7.

Then we will prove that a shock happens (and only happens) in R1. Initially, we prepare

compactly supported smooth data w(η)i (zi, 0) satisfying (4.5). Observe that after a finite time

t(η)0 (see (4.3)) these five strips are separated. Our a priori estimates are thus divided into two

parts: before and after t(η)0 .

• Estimates for t ∈ [0, t(η)0 ]: Before the separating time t

(η)0 , all the characteristic strips

Ri overlap with each other. We define

W (t) := maxi∈1,2,3,4,5,6,7

sup(x′,t′)0≤t′≤t

|wi(x′, t′)|.

In Section 4.1, via a Gronwall-type argument, we firstly obtain

W (t) = O(W(η)0 )

with W(η)0 := maxi supzi |wi(zi, 0)|. Next, we introduce the upper bounds ρi, vi inside

the strips


z′i∈[−2η,2η], 0≤s′

i≤t

ρi(z′i, s′i), S(t) := max

i∈1,2,3,4,5,6,7Si(t),

Ji(t) := sup(z′i,s′i)

z′i∈[−2η,2η] 0≤s′

i≤t

|vi(z′i, s′i)|, J(t) := maxi∈1,2,3,4,5,6,7

Ji(t).

With t(η)0 explicitly given in (4.3), it is straight forward to estimate these quantities

before t(η)0 . Integrating the transport equations under the characteristic coordinates, we

prove that when t < t(η)0

S(t) = O(1), J(t) =O(W(η)0 ).


• Estimates for t ∈ [t(η)0 ,T]: For t larger than t

(η)0 , R2 andR3 (orR5 andR6) is not com-

pletely separated due to the non-strict hyperbolicity. Nonetheless, the aforementionedfive characteristic strips R1,R2 ∪ R3,R4,R5 ∪ R6,R7 are well separated. Adapted tothis peculiarity, to replace W in this part we introduce the following quantity

W (t) := max


0≤t′≤t

w2, w3, sup(x′,t′)∈R5∪R6,

0≤t′≤t

w5, w6.

Furthermore, to give a thorough description of the dynamics after the separating time

t(η)0 , we need to estimate wi outside of the strips and we define

V1(t) := sup(x′,t′)/∈R1,

0≤t′≤t

|w1(x′, t′)|, V2(t) := sup(x′,t′)/∈R2∪R3,

0≤t′≤t

|w2(x′, t′)|, |w3(x′, t′)|,

V4(t) := sup(x′,t′)/∈R4,

0≤t′≤t

|w4(x′, t′)|, V5(t) := sup(x′,t′)/∈R5∪R6,

0≤t′≤t

|w5(x′, t′)|, |w6(x′, t′)|,

V7(t) := sup(x′,t′)/∈R7,

0≤t′≤t

|w7(x′, t′)|, V (t) := maxi=1,2,4,5,7

Vi(t).

Estimates of V before t(η)0 can be similarly obtained as we just discussed

V (t) = O(η[W(η)0 ]2).

After t(η)0 , as we emphasized in Subsection 1.2, we apply the bi-characteristic coordi-

nates to bound certain quantities. Specifically, we use this technique to estimate Vi. In

summary, for t ∈ [t(η)0 , T ], we obtain the following a priori estimates in the characteristic

strips:

S =O(1 + tV S + tJ + tSW ),

J =O(W(η)0 + tV J + tV 2S + tWJ + tV SW ),

V =O(η[W(η)0 ]2 + tV 2 + ηV J + tV W + t(W )2),

W =O(η[W(η)0 ]2 + tW 2 + tV W + tV 2).

Step 4: L∞-estimates and bootstrap argument (Subsection 4.2). We setup boot-strap assumptions in (4.59)(4.60)(4.61). Based on the above estimates in Subsection 4.1, via abootstrap argument we obtain the desired L∞-estimates of the solution:

J = O(W(η)0 ), S = O(1), V = O

(η[W

(η)0 ]2

), W = O(η[W

(η)0 ]2).

We depict the detailed behaviour of the solution up to the shock formation time. See Subsection4.2 for the details.

Step 5: Shock formation (Subsection 4.3). We shall prove that a shock forms in R1

(Theorem 1.2). With the estimates obtained in Section 4.2, we derive the following sharp controlof ρ1:

ρ1 ∼ 1− tW (η)0 .


This implies that ρ1 would become 0 (a shock forms) at a finite time T ∗η satisfying

T ∗η ∼1

W(η)0

.

Moreover, we deduce that v1 has a positive lower bound (1−ε)W (η)0 . Since v1 = ρ1w1, it follows

that w1 will blow up at T ∗η .

Step 6: H2 ill-posedness (Section 5). Furthermore, we prove the H2 ill-posedness (Theo-

rem 1.3). We first choose a family of appropriate H2-initial data (5.1)(5.4) for w(η)i in Subsection

5.1. For each η, in Subsection 5.3 we prove that the H1-norm of the solution to (1.1) tends toinfinity as t → T ∗η . Here, we apply an upper bound estimate of ∂z1ρ1 obtained in Subsection5.2. Moreover, for our solution, as η → 0, we have T ∗η → 0. The underlying reason for ill-posedness, embedded in our proof, is indeed the instantaneous shock formation. This is thedesired ill-posedness result for 3D ideal compressible MHD in H2(R3).

Finally, in Section 6 and Section 7, we prove Theorem 1.2-1.3 for the case when H1 = 0(we treat the case with non-trivial H1 in all other sections) and for the 3D compressible Eulerequations (i.e., H = 0), respectively.

1.4. Other related works. In this subsection, we refer to related works in MHD. For 3D idealcompressible MHD, Ohno-Shirota [38] proved that a characteristic fixed-boundary linearizedproblem is ill-posed in the standard Sobolev spaces H l with l ≥ 2. We also refer to [9] by Chen-Young-Zhang for a conditional5 shock formation result for 1D compressible MHD. Anotheractive topic for this system is the free-boundary problem. We refer to Chen-Wang [8], Lindblad-Zhang [32], Trakhinin-Wang [43, 44] and Zhang [53] for related results. The readers can alsoread Hu-Wang [22], Jiang-Ju-Li-Xin [24] and Li-Li [27] for other mathematical topics aboutcompressible MHD.

For incompressible MHD, the global well-posedness of Cauchy problem has been studied byCai-Lei [7], Deng-Zhang [15], He-Xu-Yu [20], Hu-Lin [21], Lin-Zhang [28], Wei-Zhang [48–50],Xu-Zhang [51]. Jeong-Oh proved ill-posedness results for the Hall- and electron-MHD in [23].For free-boundary problems, we refer to Gu-Luo-Zhang [18], Hao-Luo [19], Luo-Zhang [36],Wang-Xin [47] and references therein.

1.5. Acknowledgements. The authors would like to thank Prof. Zhouping Xin for an enlight-ening discussion.

2. Decomposition of waves

In this section, we rewrite the MHD system with the wave decomposition method. Wewill compute the eigenvalues and eigenvectors of the coefficient matrix for the decomposedsystem. We then define the corresponding characteristics and introduce the characteristic (andbi-characteristic) coordinates. With them we will trace the dynamics of characteristic waves ofdifferent travelling speeds.

5In [9], the authors considered the case of H1 = 0 and imposed the condition that the fluid density % has globallower and upper bounds.


For the 3D ideal compressible MHD (1.1), we collect the unknowns as a 8 × 1 vector-valuedfunction U with variables x1, x2, x3 and t. We let

U(x1, x2, x3, t) = (u1, u2, u3, %,H1, H2, H3, S)T (x1, x2, x3, t).

To prove ill-posedness for (1.1), as in [3, 16, 30, 31], we work under planar symmetry. Andthe solution satisfies U(x1, x2, x3, t) = U(x1, t). For notational simplicity, we denote x1 by xhereafter.

By the fifth equation of (1.1), i.e., Gauss’s Law, we obtain

∇ ·H =∂H1

∂x1+∂H2

∂x2+∂H3

∂x3=∂H1

∂x= 0, hence H1(x, t) = H1(t).

And from the third equation of (1.1), we further have dH1dt = 0. Thus, H1(x, t) equals a constant

during the evolution. We require that the constant H1 is small in the following sense

H21 minµ−1

0 Aγ, 1. (2.1)

We then collect the remaining unknowns (u1, u2, u3, %,H2, H3, S)T and they satisfy the fol-lowing equations

%∂tu1 + %u1∂xu1 + c2∂x%+ ∂Sp∂xS + µ0H2∂xH2 + µ0H3∂xH3 = 0,

%∂tu2 + %u1∂xu2 − µ0H1∂xH2 = 0,

%∂tu3 + %u1∂xu3 − µ0H1∂xH3 = 0,

∂t%+ u1∂x%+ %∂xu1 = 0,

∂tH2 + u1∂xH2 −H1∂xu2 +H2∂xu1 = 0,

∂tH3 + u1∂xH3 −H1∂xu3 +H3∂xu1 = 0,

∂tS + u1∂xS = 0.

(2.2)

Here c =√∂%p is the sound speed and the pressure p satisfies the equation of state (1.2).

We start to rewrite (2.2) as a 7× 7 hyperbolic system. Let

Φ := (u1, u2, u3, %− 1, H2, H3, S)T (x, t).

From (2.2), Φ obeys

∂tΦ +A(Φ)∂xΦ = 0 (2.3)

with the coefficient matrix

A(Φ) =

u1 0 0 c2/% µ0H2/% µ0H3/% ∂Sp/%0 u1 0 0 −µ0H1/% 0 00 0 u1 0 0 −µ0H1/% 0% 0 0 u1 0 0 0H2 −H1 0 0 u1 0 0H3 0 −H1 0 0 u1 00 0 0 0 0 0 u1

. (2.4)

The eigenvalues of (2.4) are

λ1 = u1 + Cf , λ2 = u1 + Ca, λ3 = u1 + Cs, λ4 = u1,

λ5 = u1 − Cs, λ6 = u1 − Ca, λ7 = u1 − Cf ,(2.5)


where

Cf =µ0

2%(H2

1 +H22 +H2

3 ) +c2

2+

1

2

√[µ0

%(H2

1 +H22 +H2

3 ) + c2]2 − 4µ0

%H2

1c21/2

,

Cs =µ0

2%(H2

1 +H22 +H2

3 ) +c2

2− 1

2

√[µ0

%(H2

1 +H22 +H2

3 ) + c2]2 − 4µ0

%H2

1c21/2

,

Ca =

√µ0

%H1

(2.6)

are called the fast, the slow and the Alfven wave speeds, respectively. Here, the characteristicspeed λ4 corresponds to the propagation speed of the entropy disturbance that just follows thestream-line of the flow. The pair λ2 and λ6 correspond respectively to the speeds of the Alfvendisturbances moving forward and backward along the flow. And λ3, λ5 are speeds of forwardand backward slow-magneto-acoustic disturbances, whereas λ1, λ7 constitute speeds of forwardand backward fast-magneto-acoustic disturbances. Noting that H1 satisfies (2.1), when |Φ| issmall enough, one can verify that

Cs ≤ Ca < Cf . (2.7)

The bulk of the paper is devoted to studying the case of non-trivial H1. The case of H1 = 0 willbe studied in Section 6. With non-trivial H1, by (2.7), the corresponding eigenvalues obtainedin (2.5) satisfy:

λ7 < λ6 ≤ λ5 < λ4 < λ3 ≤ λ2 < λ1. (2.8)

In (2.8), the equalities hold if and only if H22 + H2

3 = 0. Since λ2, λ3 (or λ5, λ6) could take thesame value, the system (2.3) is not strictly hyperbolic.

A very crucial step is to take proper right eigenvectors of A(Φ). By carefully designing, wetake the right eigenvectors of A(Φ) as follows:

r1 =

C2a

Cf− Cf

C2aH2

CfH1

C2aH3

CfH1

%(C2a

C2f− 1)

−H2

−H3

0

, r2 =

0CaH3H1

−CaH2H1

0−H3

H2

0

, r3 =

C2a

Cs− Cs

C2aH2

CsH1C2aH3

CsH1

%(C2a

C2s− 1)

−H2

−H3

0

, r4 =

000− %γ

001

,

r5 =

C2a

Cs− Cs

C2aH2

CsH1C2aH3

CsH1

−%(C2a

C2s− 1)

H2

H3

0

, r6 =

0CaH3H1

−CaH2H1

0H3

−H2

0

, r7 =

C2a

Cf− Cf

C2aH2

CfH1

C2aH3

CfH1

−%(C2a

C2f− 1)

H2

H3

0

.

(2.9)


As emphasized in Subsection 1.2, this selection of right eigenvectors is a key point to our proof.The corresponding left eigenvectors li are set dual to the right ones, i.e.,

lirj = δij for i, j = 1, · · · , 7. (2.10)

Here δij is the Kronecker symbol. Specifically, we list these li:

l1 =

(−

Cf2(C2

f − C2s ),

H1H2Cf (C2a − C2

s )

2(H22 +H2

3 )C2a(C2

f − C2s ),

H1H3Cf (C2a − C2

s )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

C2fC

2s

2%C2a(C2

f − C2s ),

−H2C

2f (C2

a − C2s )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

H3C2f (C2

a − C2s )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

C2fC

2s

2γC2a(C2

f − C2s )

),

l2 =

(0,

H1H3

2Ca(H22 +H2

3 ),− H1H2

2Ca(H22 +H2

3 ), 0,− H3

2(H22 +H2

3 ),

H2

2(H22 +H2

3 ), 0

),

l3 =

(Cs

2(C2f − C2

s ),−

H1H2Cs(C2a − C2

f )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

H1H3Cs(C2a − C2

f )

2(H22 +H2

3 )C2a(C2

f − C2s ),

C2fC

2s

2%C2a(C2

f − C2s ),

H2C2s (C2

a − C2f )

2(H22 +H2

3 )C2a(C2

f − C2s ),

H3C2s (C2

a − C2f )

2(H22 +H2

3 )C2a(C2

f − C2s ),

C2fC

2s

2γC2a(C2

f − C2s )

),

l4 =(0, 0, 0, 0, 0, 0, 1

), (2.11)

l5 =

(Cs

2(C2f − C2

s ),−

H1H2Cs(C2a − C2

f )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

H1H3Cs(C2a − C2

f )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

C2fC

2s

2%C2a(C2

f − C2s ),

−H2C

2s (C2

a − C2f )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

H3C2s (C2

a − C2f )

2(H22 +H2

3 )C2a(C2

f − C2s ),−

C2fC

2s

2γC2a(C2

f − C2s )

),

l6 =

(0,

H1H3

2Ca(H22 +H2

3 ),− H1H2

2Ca(H22 +H2

3 ), 0,

H3

2(H22 +H2

3 ),− H2

2(H22 +H2

3 ), 0

),

l7 =

(−

Cf2(C2

f − C2s ),

H1H2Cf (C2a − C2

s )

2(H22 +H2

3 )C2a(C2

f − C2s ),

H1H3Cf (C2a − C2

s )

2(H22 +H2

3 )C2a(C2

f − C2s ),

C2fC

2s

2%C2a(C2

f − C2s ),

H2C2f (C2

a − C2s )

2(H22 +H2

3 )C2a(C2

f − C2s ),

H3C2f (C2

a − C2s )

2(H22 +H2

3 )C2a(C2

f − C2s ),

C2fC

2s

2γC2a(C2

f − C2s )

)with Cf , Cs, Ca defined in (2.6).

Remark 2.1. Note that there is the singular factor 1H2

2+H23

present in the 2nd,3rd,5th,6th compo-

nents of all lj with j 6= 4. Nevertheless, with our carefully designed right eigenvectors (2.9), allthese 1

H22+H2

3factors are miraculously cancelled when calculating the coefficients γikm in Theorem

1.1. The details of these cancellations will be demonstrated in Section 3 and Appendix A.

We now introduce the characteristics as well as the associated characteristic and bi-characteristiccoordinates. We define the ith characteristic Ci(zi), which originates from zi, to be the image(Xi(zi, t), t) of solutions to the following ODE:

∂∂tXi(zi, t) = λi

(Φ(Xi(zi, t), t)

), t ∈ [0, T ],

Xi(zi, 0) = zi.(2.12)


For any (x, t) ∈ R× [0, T ], there is a unique (zi, si) ∈ R× [0, T ] such that (x, t) =(Xi(zi, si), si

)with Xi satisfying (2.12). We define this (zi, si) to be the characteristic coordinates of (x, t).Moreover, we define bi-characteristic coordinates: consider two transversal characteristics Ci(zi)and Cj(zj) with i 6= j. Note that the characteristic Ci(zi) can be parameterized by zj . Thus given(zi, zj) one can uniquely locate a point where the two characteristics Ci(zi) and Cj(zj) intersect.To distinguish with the characteristic coordinates above, we denote (zi, zj) by (yi, yj) in thisnew coordinate system. Then, given (x, t) ∈ R× [0, T ], the bi-characteristic coordinates of(x, t) is defined to be the unique (yi, yj) ∈ R2 such that with t = t′(yi, yj), we have

(x, t) =(Xi(yi, t


)=(Xj(yj , t


). (2.13)

To describe the compression among the ith characteristics, we introduce a geometric quantityρi (the inverse density of the ith characteristics)

ρi := ∂ziXi. (2.14)

Following from (2.12), for the initial data we have

ρi(zi, 0) = 1, (2.15)

The coordinate transformations comply with the following rules:

∂zi = ρi∂x, ∂si = λi∂x + ∂t, ∂yit′ =

ρiλj − λi

, ∂yj t′ =

ρjλi − λj

, (2.16)

∂yi =ρi

λj − λi∂sj = ∂zi +

ρiλj − λi

∂si , (2.17)

and

dx = ρidzi + λidsi, dt = dsi, dzi = dyi, dzj = dyj , (2.18)

dx =ρiλjλj − λi

dyi +ρjλiλi − λj

dyj , dt =ρi

λj − λidyi +

ρjλi − λj

dyj . (2.19)

Now we are ready to decompose system (2.3) via the decomposition of waves. See [3, 12, 25]for more applications of this method. For fixed i, setting

wi := li∂xΦ, (2.20)

and

vi := ρiwi, (2.21)

then by (2.10), we have

∂xΦ =7∑

k=1

wkrk. (2.22)

The above equation serves as a decomposition of ∂xΦ. By calculations similar to John [25], onecan verify that ρi, wi, vii=1,··· ,7 satisfy the following transport equations

∂siρi =ciiivi +(∑m 6=i

ciimwm

)ρi, (2.23)

∂siwi =− ciiiw2i +

(∑m 6=i

(−ciim + γiim)wm

)wi +

∑m 6=i,k 6=im 6=k

γikmwkwm, (2.24)


∂sivi =(∑m 6=i

γiimwm

)vi +

∑m 6=i,k 6=im6=k

γikmwkwmρi (2.25)

where ∂si = λi∂x + ∂t and

ciim = ∇Φλi · rm, (2.26)

γiim = −(λi − λm)li · (∇Φri · rm −∇Φrm · ri), m 6= i, (2.27)

γikm = −(λk − λm)li · (∇Φrk · rm), k 6= i, m 6= i. (2.28)

Remark 2.2. We call the ith characteristic genuinely nonlinear if ciii 6= 0 for all (x, t) andon the contrary linearly degenerate when ciii ≡ 0.

Remark 2.3. As will be proved in Theorem 1.1, all the coefficients ciim, γiim, γikm of the decom-posed system (2.23)-(2.25) are uniformly bounded of order 1. The proof will be given in Section3 and Appendix A. Moreover, important schematic structures of 3D compressible MHD are alsoexhibited via system (2.23)-(2.25).

3. Proof of Theorem 1.1

In this section, we explore the structures of the decomposed system (2.23)-(2.25) and aim toprove Theorem 1.1. These structures play a crucial role in proving the shock formation of (2.3).

We start from analyzing w1. Our ultimate goal is to show that w1 blows up in a finite time.Observe that in (2.24) the equation of w1 is of Ricatti-type. In particular, with the followinglemma we show that the 1st characteristic is genuinely nonlinear.

Lemma 3.1 (Genuine nonlinearity of C1). For |H1| 1 and |Φ| ≤ 2δ 1, there holdsc1

11(Φ) < 0.

Proof. Plugging the formula of r1 (2.9), λ1 (2.5), the fast and the Alfven wave speeds Cf , Ca(2.6) into the definition of c1

11(Φ) (2.26), one can verify that c111(Φ) is continuous with respect

to variable Φ and parameter H1. Since |H1| 1 and |Φ| ≤ 2δ 1, it then suffices to show thatc1

11(0) < 0 for the case of H1 = 0. With H1 = 0, it holds that

λ1 = u1 +µ0

%(H2

2 +H23 ) +Aγ%γ−1 expS

1/2. (3.1)

We hence obtain

c111(0)(0) =∇Φλ1(0) · r1(0)

=(1, 0, 0,γ − 1

2

√Aγ, 0, 0,

√Aγ

2) · (−

√Aγ, 0, 0,−1, 0, 0, 0)T

=−√Aγ(γ + 1)

2< 0.

(3.2)

Using continuity, we also conclude that c111(Φ) < 0.

Note that c111(Φ) < 0 is important for the Riccati-type equation (2.24) of w1 to develop a

potential singularity.For w2, w6, the following lemma indicates the vanishing of the Riccati-type terms.


Lemma 3.2. For cii2 and cii6 defined in (2.26), it holds that

cii2 = 0, cii6 = 0 for i = 1, · · · , 7. (3.3)

Proof. Recall the definition (2.26)

cii2 = ∇Φλi · r2.

Invoking the right eigenvector r2 in Section 2, we have

cii2 =

√µ0

%

H3

H2∂u2λi −

√µ0

%∂u3λi −

H3

H2∂H2λi + ∂H3λi. (3.4)

Since the eigenvalues λi are independent of u2 and u3, it holds that

cii2 = −H3

H2∂H2λi + ∂H3λi. (3.5)

Observe that λi in (2.5) can be written as

λi = fi(u1, ρ,H22 +H2

3 , S).

Substituting it back in (3.4), we then prove that the coefficients cii2 satisfy

cii2 = −H3

H2∂H2

2+H23fi · 2H2 + ∂H2

2+H23fi · 2H3 = 0. (3.6)

In the same fashion, we deduce cii6 = 0.

The above lemma immediately implies that c222 = 0 and c6

66 = 0. Moreover, by a straight-forward calculation, one can easily see that c4

44 = 0. Hence the 2nd, 4th, 6th characteristics arelinearly degenerate.

Remark 3.1. For ciik, γiim, γikm with other indices, we need to prove that they are O(1). Thisis achieved via conducting a heavy calculation of checking all the scenarios and exploring thedelicate cancellations among them. A 15-page detailed calculation is attached in Appendix A.

The main proposition in this section is as follows.

Proposition 3.1 (Estimates of coefficients). For |Φ| ≤ 2δ 1, the decomposed system(2.23)-(2.25) admits uniformly bounded non-trivial coefficients ciii, c

iim, γiim, γikm, i.e.,

maxi,k,m

sup|Φ|≤2δ

|ciii|, |ciim|, |γiim|, |γikm|

≤ C, (3.7)

where C is a uniform constant.

Proof. Let Γ(Φ) be the maximum of all the coefficients |ciim| and |γikm| in (2.26)-(2.28).With the eigenvalues and eigenvectors in Section 2, we will show

Γ(Φ) = O(1) for |Φ| ≤ 2δ 1. (3.8)

This is clearly true for ciim, γ44m and γ4

km as ∇Φλi, ri and l4 contain only O(1) terms. But

for γiim, γikm with i 6= 4, they might be singular since 1

H22+H2

3appears in the corresponding left

eigenvectors. Then these potentially singular coefficients could destroy all the arguments weconstruct. We must calculate all these coefficients concretely to eliminate this danger. After athorough 15-page calculation, we find that all the potentially singular denominators are cancelledin the matrix multiplication. Whence, (3.8) holds for all the coefficients (See Appendix A).


We begin with a concrete calculations for γ264. With

r2 =

0CaH3H1

−CaH2H1

0−H3

H2

0

, r4 =

000− %γ

001

, r6 =

0CaH3H1

−CaH2H1

0H3

−H2

0

and

l2 =(

0,H1H3

2Ca(H22 +H2

3 ),− H2H1

2Ca(H22 +H2

3 ), 0,− H3

2(H22 +H2

3 ),

H2

2(H22 +H2

3 ), 0),

we have

∇Φr6 · r4 =(

0,CaH3

2γH1,−CaH2

2γH1, 0, 0, 0, 0

)>,

and hence

γ264 = −(λ6 − λ4)l2 · (∇Φr6 · r4)

= −(λ6 − λ4)( H2

3

4γ(H22 +H2

3 )+

H22

4γ(H22 +H2

3 )

)(∗)= −λ6 − λ4

4γ= O(1).

Note the crucial cancellation in equality (∗).We use γ2

26 to demonstrate another type of cancellation. Proceed as above and it follows

∇Φr2 · r6 −∇Φr6 · r2 =(

0,−2CaH2

H1,−2CaH3

H1, 0, 0, 0, 0

)>.

Thus

γ226 = −(λ2 − λ6)l2 · (∇Φr2 · r6 −∇Φr6 · r2)

= (λ2 − λ6)( H3H2

H22 +H2

3

− H2H3

H22 +H2

3

)(∗∗)= 0.

Note the crucial cancellation in equality (∗∗). The computation of other coefficients, i.e., γiim, γikm

with i 6= 4, can be conducted similarly but with more involved expressions. Yet miraculouslyfor any γikm, the cancellation of singular factor 1

H22+H2

3is either in the form of (∗) or in the

form of (∗∗). The detailed calculations can be found in a 15-page Appendix A. These estimatescritically reveal the subtle structures of our system.

Now with all the uniformly bounded coefficients at our disposal, we obtain the structures ofthe system as listed in (1.5)-(1.7). This concludes the proof of Theorem 1.1.


4. Proof of shock formation (Theorem 1.2)

We prove shock formation for the 3D compressible MHD in this section. Firstly, we deduceproper a priori estimates (Subsection 4.1). These delicate estimates act as a powerful tool tocope with the strong nonlinear interaction terms. Next, employing the a priori estimates, we setup and close a bootstrap argument (Subsection 4.2). Finally, the shock formation follows fromthe L∞-estimates obtained via the bootstrap argument (Subsection 4.3).

4.1. A priori estimates. In this subsection, we derive the main a priori estimates. We firstly

define the characteristic strips and a separating time t(η)0 . Beyond t

(η)0 the strips under consid-

eration will be completely separated. The a priori estimates will also be divided into two parts:

before and after t(η)0 . These estimates provide a manifestation of how to control the nonlinear

interactions of waves with different or almost-the-same characteristic speeds.Let Ci(zi) be the ith characteristic issuing from zi with propagation speed λi. Define the

corresponding ith characteristic strip

Ri := ∪zi∈I0Ci(zi), (4.1)

where I0 = x : |x| ≤ 2η is the compact support of the initial data and η is a chosen positiveparameter. We will require η to be small to drive the ill-posedness. See Section 5.1-5.3. Yet forthe purpose of shock formation in this section, η is not necessary to be small.

For the planar symmetric 3D ideal MHD system (2.3), two pairs of characteristics are of almostthe same travelling speed. This is because H2

2 +H23 is sufficiently small, and from (2.5)(2.6) it

followsλ2 ≈ λ3, λ5 ≈ λ6.

This is illustrated by the long-time overlapping of the corresponding characteristic strips R2,R3

or R5,R6 in the picture below. Our strategy is to take unions of these overlapped strips andconsider the propagation of waves within the following five characteristic strips:

R1,R2,R4,R5,R7, (4.2)

where R2 := R2⋃R3 and R5 := R5

⋃R6. These five strips will be completely separated after

a finite separating time t(η)0 .

2η−2η t = 0

t = t(η)0

R1R2 ∪R3R4R5 ∪R6R7

Now via estimates we first determine the separating time t(η)0 . Take the supremum and

infimum of the eigenvalues

λi := supΦ∈B7

2δ(0)

λi(Φ), λi := infΦ∈B7

2δ(0)λi(Φ), for i = 1, 4, 7,


λ2 := supΦ∈B7

2δ(0)

λ2(Φ), λ3(Φ), λ2 := infΦ∈B7

2δ(0)λ2(Φ), λ3(Φ),

λ5 := supΦ∈B7

2δ(0)

λ5(Φ), λ6(Φ), λ5 := infΦ∈B7

2δ(0)λ5(Φ), λ6(Φ),

and define

σ := minα<β

α,β∈1,2,4,5,7

(λα − λβ).

According to (2.5)(2.6)(2.8), when δ is sufficiently small, σ has a uniform positive lower bound.For α ∈ 1, 2, 4, 5, 7, z ∈ I0, it follows from (2.12) that

z + λαt ≤ Xα(z, t) ≤ z + λαt.

Moreover, for all α < β, with α, β ∈ 1, 2, 4, 5, 7, there holds

Xα(−2η, t)−Xβ(2η, t) ≥ (−2η + λαt)− (2η + λβt) = −4η + (λα − λβ)t ≥ −4η + σt.

Note that the above difference is strictly positive when

t > t(η)0 :=

4η

σ> 0. (4.3)

This implies that the five characteristic strips are well separated after t(η)0 .

Let θ be a small parameter obeying 0 < θ 1. We will prescribe initial data of order O(θ)and define

W(η)0 := max

i=1,··· ,7supzi|w(η)i (zi, 0)|. (4.4)

We require the initial data satisfying the following conditions

W(η)0 = sup

z1w

(η)1 (z1, 0) = O(θ), max

i=2,3,5,6,7supzi|w(η)i (zi, 0)| ≤ Cη(W

(η)0 )2,

supz4|w(η)

4 (z4, 0)| ≤ CηW (η)0 .

(4.5)

Next, we introduce the main quantities6 to be estimated in our proof:


z′i∈[−2η,2η], 0≤s′

i≤t

ρi(z′i, s′i), S(t) := max

iSi(t), (4.6)

Ji(t) := sup(z′i,s′i)

z′i∈[−2η,2η] 0≤s′

i≤t

|vi(z′i, s′i)|, J(t) := maxiJi(t), (4.7)

W (t) := maxi

sup(x′,t′)0≤t′≤t

|wi(x′, t′)|, U(t) := sup(x′,t′)0≤t′≤t

|Φ(x′, t′)|, (4.8)

Vi(t) := sup(x′,t′)/∈Ri,

0≤t′≤t

|wi(x′, t′)| for i = 1, 4, 7,

V2(t) := max sup(x′,t′)/∈R2∪R3,

0≤t′≤t

|w2(x′, t′)|, |w3(x′, t′)|,

6For notational simplicity, we suppress the index η in the solutions.


V5(t) := max sup(x′,t′)/∈R5∪R6,

0≤t′≤t

|w5(x′, t′)|, |w6(x′, t′)|,

V (t) := maxiVi(t) for i = 1, 2, 4, 5, 7. (4.9)

Furthermore, in order to deal with the potentially harmful terms in (1.8)-(1.11), we need toestimate the following upper bound of w2, w3 (or w5, w6) in the unions of the overlapped char-acteristic strips R2 ∪R3 (or R5 ∪R6),

W (t) := max


0≤t′≤t

w2, w3, sup(x′,t′)∈R5∪R6,

0≤t′≤t

w5, w6. (4.10)

• Estimates in the region [0, t(η)0 ]: In the region t ≤ t

(η)0 , the characteristic strips Ri are

overlapped and are not separated. Nevertheless, the inverse densities of characteristics still havepositive lower bounds in this region. Thus, no shock happens. In below, for W (t), V (t), S(t),

J(t) and U(t) defined in (4.6)-(4.9), we derive a priori estimates for them before t(η)0 .

Step 1. Estimate of W (t). With t ∈ [0, t(η)0 ] by (2.24) we have

∂

∂si|wi| ≤ ΓW 2.

As in [12], via comparing wi with solutions to the following ODEddtY = ΓY 2,

Y (0) = W(η)0 ,

we obtain

|wi| ≤ Y (t) =W

(η)0

1− ΓW(η)0 t

for t < min 1

ΓW(η)0

, t(η)0

. (4.11)

And by (4.3)(3.8), we have

ΓW(η)0 t

(η)0 = O(ηW

(η)0 ) = O(θ). (4.12)

Applying (4.12) to (4.11), for a small parameter ε ∈ (0, 1100 ] it holds that

|wi(x, t)| ≤ (1 + ε)W(η)0 for any x ∈ R and t ∈ [0, t

(η)0 ].

According to (4.8), this implies that

|W (t)| ≤ (1 + ε)W(η)0 for t ∈ [0, t

(η)0 ]. (4.13)

Step 2. Estimate of S(t). From (2.23) and (3.8), ρi satisfies

∂ρi∂si

= O(ρiW ).

Integrating the above equation along Ci, we deduce

ρi(zi, t) = ρi(zi, 0) exp(O(tW (t))

). (4.14)

Then plugging (2.15) into (4.14), one can see that ρi(zi, t) is always positive. Moreover, invoking(4.3) and (4.13), it follows that

ρi(zi, t) = exp(O(ηW

(η)0 )

), for t ∈ [0, t

(η)0 ]. (4.15)


We can further choose sufficiently small θ such that

1− ε ≤ exp(O(ηW

(η)0 )

)≤ 1 + ε. (4.16)

Thus, for t ∈ [0, t(η)0 ], we have that ρi obeys the following lower and upper bounds

1− ε ≤ ρi(zi, t) ≤ 1 + ε and S(t) = O(1). (4.17)

Step 3. Estimate of V (t). First note that the initial data are compactly supported in I0.

Formulated in characteristic coordinates (z′i, s′i), this means w

(η)i (zi, 0) = 0 for z′i /∈ I0. We then

integrate (2.24) along the characteristic Ci and obtain

V (t) = O(η[W (t)]2) = O(η[W(η)0 ]2).

Step 4. Estimate of J(t). From (2.25) it follows

∂vi∂si

= O(S(t)[W (t)]2).

Using (4.13) and (4.17), for t ∈ [0, t(η)0 ], we have

J(t) = O(W(η)0 + t[W (t)]2) = O(W

(η)0 + η[W

(η)0 ]2) = O(W

(η)0 ). (4.18)

Step 5. Estimate of U(t). The estimate for U can be derived via the expression of wavedecomposition. We integrate the formula (2.22)

Φ(x, t) =

∫ x

X7(−2η,t)

∂Φ(x′, t)

∂xdx′ =

∫ x

X7(−2η,t)

∑k

wkrk(x′, t)dx′. (4.19)

Since rk(Φ) = O(1), (4.19) implies

|Φ(x, t)| = O(∑

k

∫ X1(2η,t)|

X7(−2η,t)|wk(x′, t)|dx′

). (4.20)

Employing (4.13) and the definition of U in (4.8), we get

U(t) = O(W (t)[−2η + (λ1 − λ7)t]

)= O(ηW

(η)0 ) for any t ∈ [0, t

(η)0 ].

In summary, before the separating time t(η)0 , we derive the following a priori estimates

W (t) = O(W(η)0 ), V (t) = O(η[W

(η)0 ]2), S(t) = O(1), (4.21)

J(t) = O(W(η)0 ), U(t) = O(ηW

(η)0 ).

• Estimates in the region [t(η)0 , T ]: This part encodes the key estimates of this paper. After

t(η)0 , R2 and R3 (or R5 and R6) may overlap for a long time due to the non-strict hyperbolicity.

Therefore, we let R2 := R2⋃R3 and R5 := R5

⋃R6 and investigate the well-separated five

characteristic strips R1,R2,R4,R5,R7 in this part.

Step 1. Estimate of S(t). Based on the estimates we obtain in the region [0, t(η)0 ], we move

forward to derive estimates in the region t ∈ [t(η)0 , T ] with T a potential shock formation time.


Let us start with estimating S(t), i.e., the supremum of inverse densities. For α = 1, 4, 7, if(x, t) ∈ Rα, by (2.23), we have

∂ρα∂sα

= O(Jα + V Sα). (4.22)

Thus, integrating (4.22) along the characteristic Cα, one can obtain that

Sα(t) = O(1 + tJ + tV S) with α = 1, 4, 7. (4.23)

Employing (1.5), we note that ρ3w2 is absent in the equation of ρ3 in (2.23), neither is ρ5w6 inthe equation of ρ5. Therefore, the estimate (4.23) also holds for α = 3, 5. Thus,

Sα(t) = O(1 + tJ + tV S) for α = 1, 3, 4, 5, 7. (4.24)

We then deal with ρ2. In equation (2.23)

∂ρ2

∂s2= O(J2 + V S2 + ρ2w3),

a potentially harmful nonlinear term ρ2w3 pops up. When (x, t) ∈ R2, it is possible that thepoint (x, t) also belongs to R3. So we can not use V to bound w3 directly as in (4.22). Ourstrategy is to estimate w3 in R2 ∪R3 via W . With this approach, we obtain

S2(t) = O(1 + tJ + tV S + tWS). (4.25)

Likewise, S6(t) shares the same bound as in (4.25). Together with (4.24), consequently, S(t)obeys

S(t) = O(1 + tJ + tV S + tWS). (4.26)

Step 2. Estimate of J(t). For (x, t) ∈ Rα with α = 1, 4, 7, it follows from (2.25) that

∂vα∂sα

= O(V Jα + V 2Sα).

Via integration along the characteristics, we derive the following estimates:

Jα(t) = O(W(η)0 + tV Jα + tV 2Sα) for α = 1, 4, 7. (4.27)

For vα(x, t)α=2,3,5,6 and (x, t) ∈ Rα, it holds that

∂v2

∂s2= O

(V J2 + S2V

2 + J2w3 + V S2w3

), (4.28)

∂v3

∂s3= O

(V J3 + S3V

2 + J3w2 + V S3w2

), (4.29)

∂v5

∂s5= O

(V J5 + S5V

2 + J5w6 + V S5w6

), (4.30)

∂v6

∂s6= O

(V J6 + S6V

2 + J6w5 + V S6w5

). (4.31)

In the same manner as to prove (4.25), we have

Jαα=2,3,5,6 = O(W(η)0 + tV J + tV 2S + tJW + tV SW ). (4.32)

Combining (4.27) and (4.32), we conclude that

J = O(W(η)0 + tV J + tV 2S + tJW + tV SW ). (4.33)


Step 3. Estimate of V (t). Among all a priori estimates, this step is the most delicate. Weshall utilize the bi-characteristic coordinates to cope with various situations. We firstly estimatewi(x, t) for i = 1, 4, 7 and (x, t) /∈ Ri. From (2.24), it holds for wi that

∂wi∂si

= O(V 2)+O(∑k 6=i

wk

)V+O

( ∑m 6=i,k 6=i,m 6=k

(m,k)6=(2,3),(m,k)6=(5,6)

wmwk

)+O(

(λ2 − λ3) w2w3 + (λ5 − λ6) w5w6

)︸︷︷︸

containing weak interaction

.

(4.34)

Note that Ci starts from zi /∈ I0 and terminates at (x, t) /∈ Ri. When t′ ≥ t(η)0 , every point(

Xi(zi, t′), t′

)on Ci falls either in

(R× [t

(η)0 , t]

)\⋃k=1,··· ,7Rk or in Rk for some k 6= i.

t = 0

t = t(η)0

2η−2η

RiRk (x, t)

zi

We denote Iik = t′ ∈ [t(η)0 , t] : (x, t′) ∈ Ci

⋂Rk for k 6= i. Integrating (4.34) along Ci, with

w(η)i (zi, 0) = 0, we deduce

wi(x, t) =O(tV 2 + tW 2 + V

∑k 6=i

∫ t

0wk(Xi(zi, t

′), t′)dt′)

=O(tV 2 + tW 2 + V

∑k 6=i

∫ t(η)0

0wk(Xi(zi, t

′), t′)dt′)

+O(V∑k 6=i

∫ t

t(η)0

wk(Xi(zi, t

′), t′)dt′)

=O(tV 2 + tW 2 + η[W

(η)0 ]2 + V

∑k 6=i

∫Iik

wk(Xi(zi, t

′), t′)dt′︸︷︷︸

M

).

(4.35)

For the last equality we use the fact V (t) ≤W (t) = O(W(η)0 ) for t ≤ t(η)

0 .

t = 0

t = t(η)0

2η−2η

RiRk(Xi(zi, t

′), t′)

zi yk


Now we need to bound the integral M in (4.35). When(Xi(zi, t

′), t′)∈ Iik for some k 6= i

(See the above figure), we employ the bi-characteristic coordinates to estimate the integral M∫Iik

wk(Xi(zi, t

′), t′)dt′

=O(∫

yk∈[−2η,2η]

∣∣∣∣ρk(yk, t

′(yi, yk))

λi − λkwk(yk, t

′(yi, yk))∣∣∣∣dyk)

=O(ηJk).

(4.36)

Here we use the fact that λi − λk 6= 0 for i = 1, 4, 7. Then, together with (4.35) and (4.36), fori = 1, 4, 7, we obtain

wi(x, t) = O(η[W(η)0 ]2 + tV 2 + tW 2 + ηV J) for any (x, t) /∈ Ri. (4.37)

Next, we estimate other wi, i.e., Vii=2,3,5,6. Here we provide the details of the estimate forV3. Vjj=2,5,6 can be handled in a similar way. For w3(x, t), with (x, t) /∈ R2 ∪R3, we have

∂w3

∂s3= O

(V 2 +

( ∑k 6=2,k 6=3

wk

)V +

∑m,k 6=3,m 6=k,(m,k)6=(5,6)

wmwk + (λ5 − λ6)w5w6

). (4.38)

Integrating ∂s3w3 along C3, it is possible that some points (x′, t′) ∈ C3 lie in the region R5 ∪R6

for t(η)0 ≤ t′ < t. Taking into account of this scenario, we derive the following estimate

w3(x, t) =O(tV 2 + tV W + tW 2 + V

∑k=1,4,7

∫ t

0wk(X2(z2, t

′), t′)dt′)

=O(tV 2 + η[W

(η)0 ]2 + V

∑k=1,4,7

∫I3k

wk(X2(z2, t

′), t′)dt′).

(4.39)

Similar to (4.35) and (4.36), it then follows

V3 = O(η[W

(η)0 ]2 + tV 2 + tV W + tW 2 + ηV J

). (4.40)

Thus, we conclude from (4.37) and (4.40)

V = O(η[W

(η)0 ]2 + tV 2 + tV W + tW 2 + ηV J

). (4.41)

Step 4. Estimate of W (t). Now we proceed to estimate

W (t) = max


0≤t′≤t

w2, w3, sup(x′,t′)∈R5∪R6,

0≤t′≤t

w5, w6,

i.e., the supremum of w2, w3 (or w5, w6) taken in the unions of the corresponding overlappedcharacteristic strips R2∪R3 (or R5∪R6). Note that the strong interaction of w2, w3 (or w5, w6)will appear in the following estimates. And their characteristic speeds are almost-the-same. Wefocus on estimating w3(x, t) with (x, t) ∈ R2 ∪R3. Recall the equation of w3,

∂w3

∂s3= O

[w2

3 + w2w3 + w2

( ∑k 6=2,k 6=3

wk

)+ w3

( ∑k 6=2,k 6=3

wk

)+

∑m,k 6=2,3,m 6=k

wmwk

]. (4.42)


The marked term w2w3 corresponds to the strong nonlinear interaction. Integrating (4.42), itholds that

w3(x, t) = O(η[W

(η)0 ]2 + tW 2 + tV W + tV 2

), for (x, t) ∈ R2 ∪R3. (4.43)

The estimates for w2, w5, w6 can be obtained in a same manner as for w3. And hence we prove

W (t) = O(η[W(η)0 ]2 + tW 2 + tV W + tV 2). (4.44)

Step 5. Estimate of U(t). Finally, the estimate of Φ can be deduced via the aforementionedestimates. If (x, t) does not belong to any characteristic strip, one can go back to (4.20) andobtain

U(t) = O((−2η + (λ1 − λ7)t)V

). (4.45)

If (x, t) ∈ Rk for some k, we have

|Φ(x, t)| =∣∣∣ ∫ x

Xk(−2η,t)

∂Φ(x′, t)

∂xdx′∣∣∣ =

∣∣∣ ∫ x

Xk(−2η,t)

∑k

wkrk(x′, t)dx′

∣∣∣≤∫ Xk(2η,t)

Xk(−2η,t)|wk(x′, t)|dx′ = O

(∫ 2η

−2η|wk(zk, t)|ρkdzk

)= O(ηJ).

(4.46)

Combining (4.45) and (4.46), we therefore obtain

U(t) = O(ηJ + ηV + tV ), for any t ∈ [t(η)0 , T ]. (4.47)

In conclusion, for all t ∈ [0, T ], we derive the following a priori estimates

S =O(1 + tV S + tJ + tSW ), (4.48)

J =O(W(η)0 + tV J + tV 2S + tWJ + tV SW ), (4.49)

V =O(η[W(η)0 ]2 + tV 2 + ηV J + tV W + tW 2), (4.50)

W =O(η[W(η)0 ]2 + tW 2 + tV W + tV 2), (4.51)

U =O(ηJ + ηV + tV ). (4.52)

These a priori estimates will be used in a bootstrap argument in the next subsection. Afterclosing the bootstrap argument, we will obtain the upper bounds for all the quantities definedin (4.6)-(4.10).

4.2. L∞-estimates and bootstrap argument. In this subsection, we design a bootstrapargument to bound S, J , V , W and U . More precisely, employing the desired a priori estimates(4.48)-(4.52), we obtain the following proposition:

Proposition 4.1 (L∞-estimates). For t ∈ [0, T ∗η ) with T ∗η being the (hypothetical) shock for-mation time satisfying

T ∗η ≤C

W(η)0

, (4.53)

we have

S(t) =O(1), (4.54)

J(t) =O(W(η)0 ), (4.55)


V (t) =O(η[W

(η)0 ]2

), (4.56)

W (t) =O(η[W

(η)0 ]2

), (4.57)

U(t) =O(W(η)0 ). (4.58)

Remark 4.1. A posteriori, we will prove that the shock formation time T ∗η verifies (4.53) inthe next Subsection 4.3. In (4.78), we will also specify the value of the constant C in (4.53).

Proof. When t = 0, by (2.15) and definitions in (4.6)-(4.10), it holds

S(0) = 1, J(0) = W(η)0 , V (0) = 0.

To obtain (4.54)-(4.56), with θ small, we impose bootstrap assumptions:

tV ≤θ12 , (4.59)

tW ≤θ12 , (4.60)

J ≤θ12 . (4.61)

Applying (4.59) and (4.60) to (4.48), we have

S = O(1 + tJ + θ12S) when θ is small, this implies S = O(1 + tJ). (4.62)

To control J(t), we appeal to (4.49). Using (4.59)(4.60)(4.62), we obtain

via plugging (4.59)(4.60) into (4.49), J = O(W(η)0 + θ

12J + θ

12V S).

Absorb θ12J , we have J = O(W

(η)0 + θ

12V S)

by (4.62), = O(W(η)0 + θ

12V + θJ).

Absorb θJ , it holds J = O(W(η)0 + θ

12V ). (4.63)

We proceed to estimate V (t). Applying (4.59)-(4.61) to (4.50), we get

V = O(η[W

(η)0 ]2 + θ

12V + ηθ

12V + θ

12 W).

For 0 < θ 1, the second and third terms on the right can be absorbed by the left, hence itinfers

V = O(η[W

(η)0 ]2 + θ

12 W). (4.64)

We then deal with W , invoking (4.59)(4.60)(4.64) into (4.51), we deduce

via plugging (4.59)(4.60) into (4.51), W = O(η(W(η)0 )2 + θ

12 W + θ

12V ).

Absorb θ12 W , we have W = O(η(W

(η)0 )2 + θ

12V )

by (4.64), = O(η[W(η)0 ]2 + θ

12 η(W

(η)0 )2 + θW ).

Absorb θW , it holds W = O(η(W(η)0 )2). (4.65)

This finishes the proof of (4.57). We then substitute (4.65) into (4.64), hence (4.56) can beverified. Furthermore, with (4.53), we have

tV = O(ηW(η)0 ) = O(θ) < θ

12 , (4.66)


tW = O(ηW(η)0 ) = O(θ) < θ

12 . (4.67)

Note that (4.66) and (4.67) improve the bootstrap assumptions (4.59) and (4.60).Next, for J(t) and S(t), it follows from (4.63)(4.56) that the estimates (4.55) and (4.54) are

valid:

J = O(W(η)0 + θ

12 η[W

(η)0 ]2) = O(W

(η)0 ) = O(θ) < θ

12 , (4.68)

S = O(1 + tW(η)0 ) = O(1). (4.69)

And (4.68) improves the bootstrap assumption (4.61).Finally, by (4.54)-(4.57) we deduce the estimate for U

U(t) = O(ηW(η)0 + η2W

(η)0 + η[W

(η)0 ]2) = O(ηW

(η)0 ) = O(θ), t ∈ [0, T ∗η ]. (4.70)

Furthermore, since U is the supremum of Φ as defined in (4.8), with sufficiently small θ, we thenconclude that the solution Φ always stays in B7

δ (0).

4.3. Shock formation. We will finish the proof of Theorem 1.2 in this subsection. In particular,we will show that in R1 the inverse density ρ1 → 0 and w1 →∞ as t→ T ∗η while Φ is uniformlybounded. Hence a shock happens at T ∗η .

Proof of Theorem 1.2. Our first step is to control v1. We integrate (2.25)

∂v1

∂s1= O

(∑m 6=1

wm

)v1 + O

( ∑m,k 6=1,m6=k

wmwk

)︸︷︷︸containing weak interaction

ρ1

along C1 and obtain

v1(z1, t) = w(η)1 (z1, 0) +O(tV J + tV 2S) = w

(η)1 (z1, 0) +O(η[W

(η)0 ]2). (4.71)

In particular, taking z0 ∈ I0 such that w(η)1 (z0, 0) = W

(η)0 , then for sufficiently small θ, the above

formula yields

(1− ε)W (η)0 ≤ v1(z0, t) ≤ (1 + ε)W

(η)0 . (4.72)

We then estimate ρ1. Using (2.23)

∂ρ1

∂s1= c1

11(Φ)v1 +O(∑k 6=1

wk

)ρ1,

and the fact c111(Φ) < 0, we obtain

− c111(Φ)v1 −

∣∣∣O(∑k 6=1

wk

)∣∣∣ρ1 ≤∂ρ1

∂s1≤ −c1

11(Φ)v1 +∣∣∣O(∑

k 6=1

wk

)∣∣∣ρ1. (4.73)

We will then employ the Gronwall’s inequality to derive upper and lower bounds of ρ1. Firstly,

by (4.70), we have |Φ| = O(ηW(η)0 ) = O(θ) ≤ δ. Setting θ to be sufficiently small, we further

have

(1− ε)|c111(0)| ≤ −c1

11(Φ) ≤ (1 + ε)|c111(0)|. (4.74)


For the coefficient O(∑

k 6=1wk

)in (4.73), using bi-characteristic coordinates (as in Subsection

4.2), we deduce∫ t

0

∑k 6=1

|wk(X1(z1, t′), t′)|dt′ =

∑k 6=1

∫ 2η

−2η

|vk|λ1 − λk

dyk = O(ηW(η)0 + ηJ) = O(ηW

(η)0 ) = O(θ).

Here we use the fact that the factor 1λ1−λk arising from the bi-characteristic transformation is

of O(1). For 0 < θ 1, the above estimate implies

1− ε ≤ exp(∫ t

0±|O

(∑k 6=1

wk(X1(z1, t′), t′)

)|dt′)≤ 1 + ε. (4.75)

Applying the Gronwall’s inequality to (4.73) and combining with (4.74)-(4.75), we get

(1− ε)(1− (1 + ε)2|c111(0)|

∫ t

0v1(z1, t

′)dt′)

≤ρ1(z1, t)

≤(1 + ε)(1− (1− ε)2|c111(0)|

∫ t

0v1(z1, t

′)dt′).

(4.76)

By the first inequality of (4.72), we then arrive at

ρ1(z0, t) ≥ (1− ε)(

1− (1 + ε)3|c111(0)|tW (η)

0

), (4.77)

which shows that ρ1(z0, t) > 0 when t < 1

(1+ε)3|c111(0)|W (η)0

. Meanwhile, the second inequality of

(4.72) implies

ρ1(z0, t) ≤ (1 + ε)(

1− (1− ε)3|c111(0)|tW (η)

0

).

Together with (4.77), we conclude that there exists a finite positive time T ∗η (shock formationtime) such that

limt→T ∗η

ρ1(z0, t) = 0 and limt→T ∗η

w1(z0, t) = limt→T ∗η

v1

ρ1(z0, t) =∞.

And T ∗η obeys1

(1 + ε)3|c111(0)|W (η)

0

≤ T ∗η ≤1

(1− ε)3|c111(0)|W (η)

0

. (4.78)

As we mentioned in Remark 4.1, this verifies the condition (4.53).

Recall that the estimates of V (t) and W (t) in Section 4.2 show that the unknowns wi(z, t)i=2,3,5,6

are bounded in the whole space before the shock formation time T ∗η , and wi(z, t)i=4,7 arebounded when (z, t) /∈ Rii=4,7. In fact, now we prove that wi(z, t)i=4,7 are also regular inRii=4,7. To show this, we introduce

W ′i (t) := sup(z,t′)∈Ri0≤t′≤t

wi(z, t′) for i = 4, 7 (4.79)

and prove the following:


Proposition 4.2. For t ∈ [0, T ∗η ) with the shock formation time T ∗η satisfying (4.78), we have

W ′i (t) = O(W(η)0 ).

Proof. We study w4 and w7 separately. For w4, since c444 = 0, it verifies

∂w4

∂s4= O

(∑k 6=4

wk

)w4 + O

( ∑m 6=4,k 6=4m 6=k

wmwk

)︸︷︷︸


. (4.80)

Let (z, t) ∈ R4 and t ∈ [t(η)0 , T ∗η ], integrating (4.80) along C4, we obtain

w4(z, t) ≤ O(W(η)0 + tV W ′4 + tV 2). (4.81)

In view of the facts tV = O(θ) and V = O(η[W

(η)0 ]2

), we hence get

W ′4 ≤ O(W(η)0 ). (4.82)

For w7, even though its evolution equation is of Riccati-type, its initial datum is designed tobe much more smaller than that of w1. This guarantees that w7(z, t) is regular when t ≤ T ∗η .Invoking the estimate of V , it holds that

∂w7

∂s7=− c7

77w27 +O

(∑k 6=7

wk

)w7 + O

( ∑m 6=7,k 6=7m 6=k

wmwk

)︸︷︷︸


=− c777w

27 +O(V )w7 +O(V 2)

=− c777w

27 +O

(η[W

(η)0 ]2

)w7 +O

(η2[W

(η)0 ]4

).

(4.83)

Equivalently,

d

ds

[exp

(O(η[W

(η)0 ]2

)s)w7

]= − exp

(O(η[W

(η)0 ]2

)s)c777w

27 + exp

(O(η[W

(η)0 ]2

)s)O(η2[W

(η)0 ]4

). (4.84)

Since

exp(O(η[W

(η)0 ]2

)s)≤ exp

(O(η[W

(η)0 ]2

)T ∗η

)≤ exp

(O(ηW

(η)0

))= O(eθ), (4.85)

by choosing θ sufficiently small, we have

1− ε ≤ exp(O(η[W

(η)0 ]2

)s)≤ 1 + ε. (4.86)

Thus, back to (4.84), it holds

d

ds

[exp

(O(η[W

(η)0 ]2

)s)w7

]≤ Cw2

7 +O(η2[W

(η)0 ]4

). (4.87)

Integrating (4.87) along C7, we obtain

W ′7 ≤ O(w7(z, 0) + tW ′27 + tη2[W

(η)0 ]4

)≤ O

([W

(η)0 ]2 + tW ′27 + η2[W

(η)0 ]3

)≤ O

([W

(η)0 ]2 + tW ′27

).

(4.88)


Now we introduce an additional bootstrap assumption

tW ′7 ≤ θ12 . (4.89)

Then by (4.88), it holds

W ′7 ≤ O([W

(η)0 ]2

)(4.90)

and the bootstrap assumption (4.89) can be improved by

tW ′7 ≤ O(W(η)0 ) = O(θ) < θ

12 . (4.91)

Moreover, we show that the shock formation only occurs in R1 by deriving positive lowerbounds for ρ2, · · · , ρ7. This concludes the proof of Theorem 1.2.

Proposition 4.3. For t ≤ T ∗η , we have ρii=2,··· ,7 are all bounded away from zero in the whole(x, t)-plane, i.e., ρi > C > 0 for a uniform constant C.

Proof. To derive a uniform lower bound for ρi, we start from estimates inside Ri. Similarly to(4.76), via estimating (2.23), we have

ρi(zi, t) ≥ (1− ε)(

1− (1 + ε)2|ciii(0)|∫ t

0|vi(zi, t′)|dt′

).

Employing the estimates of vi, one can then bound ρi from below in Ri.We then prove that ρii=1,··· ,7 obey a positive lower bound outside Ri. For the equation

(2.23) of ρi, applying the Gronwall’s inequality and estimates obtained in Section 4, one can

obtain that for any zi /∈ [η, 2η], t ∈ [t(η)0 , T ∗η ),

ρi(zi, t) ≥ exp(−∫ t

0

∣∣∣O(∑k

wk(zi, s))∣∣∣ds) ≥ 1− ε. (4.92)

Interested readers are also referred to Proposition 8.1 in [3] for a more detailed discussion.

5. Proof of H2 ill-posedness (Theorem 1.3)

In this section, we use the above shock formation to prove H2 ill-posedness for 3D ideal MHD

equations (1.1). We first construct a family of initial data w(η)i (z, 0), which will lead to the

ill-posedness. For each η, we will prove that there is a corresponding shock formation at finitetime T ∗η . The meaning of the ill-posedness is twofold: Firstly, the lifespan T ∗η will vanish when

η → 0. Moreover, the L2 norm of w1 will blow up at T ∗η .

5.1. Initial data designed for proving ill-posedness. Our first step is to construct theinitial data. Counterexamples of local well-posedness will evolve from them.

We design a family of initial data w(η)1 (z, 0) satisfying

w(η)1 (z, 0) = θ

∫Rζ η

10(y)| ln(z − y)|αχ(z − y)dy with 0 < α <

1

2. (5.1)

Here, η > 0 is a small parameter and χ is the characteristic function

χ(z) =

1, z ∈ [6

5η,95η],

0, z /∈ (65η,

95η),


and ζ η10

(z) is a test function in C∞0 (R) verifying

supp ζ η10

(z) = z : |z| ≤ η

10, and

∫Rζ η

10(z)dz = 1.

For such initial datum, we present here the estimate of its H1 norm in the following lemma.

Lemma 5.1. Let 0 < α < 12 .We construct a function w1 as below.

w1(x, x2, x3) = w(η)1 (x, 0) := θ

∫Rζ η

10(y)| ln(x− y)|αχ(x− y)dy in B3

η2, (5.2)

where B3η2

denotes the 3D unit ball of radius η2 centered at the origin. Then, we have that

w1 ∈ H1(R3) satisfying

‖w1‖H1(R3) . θ√η(1 + | ln 6η

5|α + | ln 9η

5|α). (5.3)

Proof. See Lemma A.1 in [3] for a detailed similar calculation.

For k = 2, · · · , 7, choose

w(η)k (z, 0) = θ2

∫Rζ η

10(y)χ(z − y)dy. (5.4)

One can verify that these initial data satisfy all the requirements in Section 4.Recall that

Φx(x, t) =

7∑k=1

wk(x, t)rk(Φ). (5.5)

Now we go back to the original system (2.3) by reversing the process of wave decomposition.

The initial data of the original variables Φ are denoted as Φ(x, 0) = Φ0(x, x2, x3). According to(5.5), the initial data satisfy

∂xΦ0(x, x2, x3) =7∑

k=1

wk(x, x2, x3)rk(Φ(x, 0)

). (5.6)

Since rk constructed in (2.9) is Lipschitz continuous in Φ, by a standard ODE argument, given

the aforementioned wk(x, x2, x3), one can solve the ODE system (5.6) for Φ0. Moreover, because

rk(Φ) ∈ L∞(B7δ (0)) and wk ∈ H1(R3), we have Φ0 ∈ H2(R3). These constitute the H2 initial

data for the 3D ideal compressible MHD equations (2.3).

5.2. Estimate for ∂z1ρ1. In this subsection, we derive a uniform bound for |∂z1ρ1|. As we willsee in Subsection 5.3, this upper bound plays a crucial role in proving the desired ill-posednessresult.

For a fixed η > 0, we employ characteristic and bi-characteristic coordinates introduced in(2.12)-(2.14) for i 6= j:

(x, t) =(Xi(zi, si), si

)=(Xi(yi, t


)=(Xj(yj , t


).

As discussed in Section 2, the transformations between these coordinates satisfy

∂yi =ρi

λj − λi∂sj = ∂zi +

ρiλj − λi

∂si . (5.7)


Along C1 we fix y1 and set y7 as a parameter. From (5.7), we have

∂y1 = ∂z1 +ρ1

λ7 − λ1∂s1 = ∂z1 −

ρ1

2Cf∂s1 , (5.8)

where Cf is the fast wave speed defined in (2.6). We now state the main result of this subsection.

Proposition 5.1. Fix η, for all zi ∈ R and any t < T ∗, there holds |∂z1ρ1| ≤ Cη for a positiveconstant Cη.

Proof. Using (5.8), we get

∂z1ρ1 = ∂y1ρ1 +ρ1

2Cf∂s1ρ1. (5.9)

To bound ∂z1ρ1, we start with controlling ∂y1ρ1. Let

τ(7)1 := ∂y1ρ1, π

(7)1 := ∂y1v1.

Since

∂y7 =ρ7

λ1 − λ7∂s1 =

ρ7

2Cf∂s1 , (5.10)

we have that τ(7)1 obeys

∂y7τ(7)1 =∂y1∂y7ρ1 = ∂y1

( ρ7

2Cf∂s1ρ1

)= ∂y1

(ρ1ρ7

2Cf

∑m

c11mwm

)=ρ7

2Cf

(c1

11∂y1v1 +∑m6=1

c11mwm∂y1ρ1

)+

ρ1

2Cf

(∑m6=7

c11mwm∂y1ρ7 + c1

17∂y1v7

)+ρ1ρ7

2Cf

( ∑m 6=1,m 6=7

c11m

1

ρm∂y1vm −

∑m 6=1,m 6=7

c11m

wmρm

∂y1ρm

)−∂y1Cf2C2

f

ρ1ρ7

∑m

c11mwm +

ρ1ρ7

2Cf

∑m

∂y1c11mwm.

(5.11)

We still need to estimate ∂y1Cf , ∂y1c11m, ∂y1ρm and ∂y1vm in (5.11). Note that for Cf , its deriv-

ative satisfies

∂y1Cf =∇Φ(λ1 − u1) · ∂y1Φ = ∇Φλ1 · [∂s7X7∂y1t′∂xΦ + ∂y1t

′∂tΦ]

=∇Φ(λ1 − u1) ·[λ7

ρ1

λ7 − λ1

∑k

wkrk +ρ1

λ7 − λ1(−A(Φ)

∑k

wkrk)]

=O(v1 + ρ1

∑k 6=1

wk

).

(5.12)

Similarly, for c11m, we also have

∂y1c11m =∇Φc

11m · ∂y1Φ = ∇Φc

11m · [∂s7X7∂y1t

′∂xΦ + ∂y1t′∂tΦ]

=∇Φc11m ·

[λ7

ρ1

λ7 − λ1

∑k

wkrk +ρ1

λ7 − λ1(−A(Φ)

∑k

wkrk)]

=O(v1 + ρ1

∑k 6=1

wk

).

(5.13)


By (5.7), it holds

∂y1ρm =ρ1

λm − λ1∂smρm = O

(ρmv1 + ρ1ρm

∑k 6=1

wk

)when m 6= 1, (5.14)

and

∂y1vm =ρ1

λm − λ1∂smvm = O

(ρmv1

∑k 6=1

wk + ρ1ρm∑

j 6=1,k 6=1j 6=k

wjwk

)when m 6= 1. (5.15)

Together with the estimates obtained in Proposition 4.1, (5.12)-(5.15) imply that (5.11) couldbe rewritten as

∂y7τ(7)1 = Bη

11τ(7)1 +Bη

12π(7)1 +Bη

13, (5.16)

where Bη11, B

η12, B

η13 are uniformly bounded constants depending on η.

In the same fashion, we obtain the evolution equation for π(7)1 :

∂y7π(7)1 =

ρ7

2Cf

( ∑p 6=1,q 6=1p6=q

γ1pqwpwqτ

(7)1 +

∑p 6=1

γ11pwpπ

(7)1

)

−∂y1Cf4C2

f

(∑p 6=1

γ11pwpv1ρ7 +

∑p 6=1,q 6=1p6=q

γ1pqwpwqρ7ρ1

)+ρ7ρ1

2Cf

(∑p 6=1

∂y1γ11pwpwm +

∑p 6=1,q 6=1p 6=q

∂y1γ1pqwpwq

)+

ρ1

2Cf

(∑p 6=1

γ11pwpw1 +

∑p6=1,q 6=1p6=q

γ1pqwpwq

)∂y1ρ7

+ρ7ρ1

2Cf

( ∑p 6=1,p 6=7

γ11p

w1

ρp+

∑p 6=1,q 6=1p6=q

γ1pq

wqρp

)(∂y1vp − wp∂y1ρp)

+ρ7ρ1

2Cf

∑p 6=1,q 6=1p6=q

γ1pq

wpρq

(∂y1vq − wq∂y1ρq) +ρ1

2Cfγ1

17w1∂y1v7

=Bη21τ

(7)1 +Bη

22π(7)1 +Bη

23,

(5.17)

where one can check as in (5.16) Bη21, B

η22, B

η23 are also uniformly bounded.

Next, we check the initial data of τ(7)1 and π

(7)m . Since ρ1(z1, 0) = 1, for fixed η, it follows from

(5.7) that

τ(7)1 (z1, 0) =∂z1ρ1(z1, 0)− ρ1(z1, 0)

2Cf∂s1ρ1(z1, 0)

=− 1

2Cf

∑k

c11kw

(η)k (z1, 0) = O(W

(η)0 ) < +∞.


And with v1(z1, 0) = w(η)1 (z1, 0), we similarly deduce

π(7)1 (z1, 0) =∂z1v1(z1, 0)− ρ1(z1, 0)

2Cf∂s1v1(z1, 0)

=∂z1w(η)1 (z1, 0)− 1

2Cf

∑q 6=1,q 6=p

γ1pqwp(z1, 0)wq(z1, 0)ρ1(z1, 0)

=O(∂z1w(η)1 (z1, 0) + [W

(η)0 ]2) < +∞.

Applying Gronwall’s inequality to (5.16) and (5.17), for t ≤ T ∗η we deduce that τ(7)1 := ∂y1ρ1 is

bounded by a constant depending on η.Back to (5.9), we hence prove

∂z1ρ1 = ∂y1ρ1 +O(v1 +∑m 6=1

wmρ1).

With the bounds for J(t), S(t) and V (t) in Section 4.2, consequently, we arrive at the conclusionthat ∂z1ρ1 is bounded by Cη, which is a constant depending on η. Note that the dependence onη causes no trouble since the blow-up is derived for each fixed η in Subsection 5.3.

5.3. Ill-posedness mechanism. We prove our main result Theorem 1.3 in this section. Forour proof, one can see that the ill-posedness is driven by the shock formation.

Proof of Theorem 1.3. First, the ill-posedness builds on an instantaneous shock formation. Moreprecisely, for each η the solution yields a shock at a finite time T ∗η and T ∗η → 0 as η → 0. Withinitial data (5.1) and (5.4), employing the estimates in Subsection 4.1 and Subsection 4.2, weknow that a shock forms in R1 at time T ∗η with

1

(1 + ε)3|c111(0)|W (η)

0

≤ T ∗η ≤1

(1− ε)3|c111(0)|W (η)

0

. (5.18)

By (5.1), we see that w(η)1 (z, 0) = θ · O(| ln z|α) with 0 < α < 1

2 . As η → 0, the support of the

initial data shrinks to the origin and hence |w(η)1 (z, 0)| → ∞. It then follows from the definition

of W(η)0 in (4.4) and the above estimate (5.18) that T ∗η → 0.

Furthermore, we will show the H1-norm of solutions to (1.1) blows up at t = T ∗η . In particular,

we give a lower bound estimate of the H2-norm in the following spatial region

ΩT ∗η = (x, x2, x3, T∗η ) : x = X1(z, T ∗η ), (x−λ1T

∗η −

3η

2)2+x2

2+x23 ≤

η2

4, for η ≤ z ≤ 2η. (5.19)

To calculate∫

ΩT∗η|w1(x, x2, x3, T

∗η )|2dxdx2dx3, we first compute the area element

∫dx2dx3.

By the definition of characteristics, for a characteristic line C1 starting from z, its terminal attime T ∗η is x = z + λ1T

∗η . Thus, one can verify that the area of ΩT ∗η is∫

x22+x2

3≤η2

4−(x−λ1T ∗η−

3η2

)2

dx2dx3 =π[η2

4− (z − 3η

2)2]

= π(z − η)(2η − z). (5.20)


Employing characteristic coordinates, we obtain

‖w1‖2L2(ΩT∗η ) = π

∫ η

η2

∣∣v1

ρ1(z, T ∗η )

∣∣2ρ1(z, T ∗η )(z − η)(2η − z)dz. (5.21)

Recall the estimate of v1 in (4.71)

v1(z, t) = w(η)1 (z, 0) +O(η[W

(η)0 ]2). (5.22)

We integrate in a subinterval (z0, z∗0 ] ⊆ [η, 2η] where |w(η)

1 (z, 0)| > 12W

(η)0 for z ∈ (z0, z

∗0 ]. Then

in this subinterval, it follows from (5.22) that |v1(z, t)| admits a lower bound |v1(z, t)| ≥ 14W

(η)0 .

Applying this in (5.21) and using the fact that ρ1(z0, T∗η ) = 0, we get

‖w1‖2L2(ΩT∗η ) ≥ C∫ z∗0

z0

∣∣v1

ρ1(z, T ∗η )

∣∣2ρ1(z, T ∗η )(z − η)(2η − z)dz

≥ C(W(η)0 )2(z0 − η)(2η − z∗0)

∫ z∗0

z0

1

ρ1(z, T ∗η )− ρ1(z0, T ∗η )dz︸︷︷︸

by ρ1(z0, T ∗η ) = 0

≥ C(W(η)0 )2(z0 − η)(2η − z∗0)

∫ z∗0

z0

1

(z − z0) sup |∂zρ1|dz︸︷︷︸

by mean value theorem

≥ Cη∫ z∗0

z0

1

z − z0dz = +∞.

(5.23)

This shows the blow-up of the L2-norm of w1. In (5.23), we crucially use the fact that |∂zρ1| isbounded from Proposition 5.1.

The last step is going back to the original physical unknowns Φ and show the blow-up of‖Φ‖H1 . Due to the boundedness of sup(x,T ∗η )∈Ri wi(x, T

∗η )i=2,··· ,7 in (4.65), (4.82) and (4.90),

it holds sup

(x,T ∗η )∈Riwi(x, T

∗η )i=2,··· ,7

≤ CW (η)0 , (5.24)

together with the fact that the first components of r2, r4 and r6 are zero, we deduce

‖∂xu1‖L2(ΩT∗η ) =‖7∑

k=1

wkr1k‖L2(ΩT∗η ) ≥ C

(‖w1‖L2(ΩT∗η ) −

∑k=3,5,7

‖wk‖L2(ΩT∗η )

)≥C(‖w1‖L2(ΩT∗η ) − 3η

32W

(η)0

)≥ +∞.

Here rik denotes the ith component of the right eigenvector rk. Hence the blow-up of ‖w1‖L2(ΩT∗η )

implies that ‖u1‖H1(ΩT∗η ) is also infinite. Similarly, we can show ‖∂x%‖L2(ΩT∗η ) = +∞.

‖∂x%‖L2(ΩT∗η ) =‖7∑

k=1

wkr4k‖L2(ΩT∗η ) ≥ C

(‖w1‖L2(ΩT∗η ) −

∑k=3,4,5,7

‖wk‖L2(ΩT∗η )

)≥C(‖w1‖L2(ΩT∗η ) − 4η

32W

(η)0

)≥ +∞.


This finishes the proof of Theorem 1.3.

6. Case of H1 = 0

In this section, we explain how to deal with the case when H1 = 0. We analyze the detailedsubtle structures of the system (1.4) under this scenario. And we explain the steps to proveshock formation and ill-posedness when H1 = 0.

With H1 = 0, Φ also satisfies (2.3):

∂tΦ +A(Φ)∂xΦ = 0,

but with a different coefficient matrix, which is

A(Φ) =

u1 0 0 c2/% µ0H2/% µ0H3/% ∂Sp/%0 u1 0 0 0 0 00 0 u1 0 0 0 0% 0 0 u1 0 0 0H2 0 0 0 u1 0 0H3 0 0 0 0 u1 00 0 0 0 0 0 u1

. (6.1)


λ1 = u1 + Cf , λ2 = λ3 = λ4 = λ5 = λ6 = u1, λ7 = u1 − Cf . (6.2)

Here, Cf denotes the fast wave speed. It is defined as

Cf =µ0

%(H2

2 +H23 ) + c2

1/2.

And the slow and Alfven wave speeds Cs, Ca vanish. The eigenvalues for A(Φ) in (6.1) satisfy

λ7 < λ6 = λ5 = λ4 = λ3 = λ2 < λ1. (6.3)

There exist seven linearly independent right eigenvectors:

r1 =

−100

−%/Cf−H2/Cf−H3/Cf

0

, r2 =

0100000

, r3 =

0010000

, r4 =

000

−∂Sp/c2

001

,

r5 =

000

−µ0H2/c2

100

, r6 =

000

−µ0H3/c2

010

, r7 =

−100

%/CfH2/CfH3/Cf

0

.

(6.4)


In view of (6.2)-(6.4), when H1 = 0, our system is a non-strictly hyperbolic system with aquintuple eigenvalue λ2 = λ3 = λ4 = λ5 = λ6 = u1. The dual left eigenvectors are

l1 =(− 1

2, 0, 0,− c2

2%Cf,−µ0H2

2%Cf,−µ0H3

2%Cf,− ∂Sp

2%Cf

),

l2 = (0, 1, 0, 0, 0, 0, 0), l3 = (0, 0, 1, 0, 0, 0, 0), l4 = (0, 0, 0, 0, 0, 0, 1),

l5 =(

0, 0, 0,−c2H2

%C2f

,µ0H

23 + %c2

%C2f

,−µ0H2H3

%C2f

,−H2∂Sp

%C2f

),

l6 =(

0, 0, 0,−c2H3

%C2f

,−µ0H2H3

%C2f

,µ0H

22 + %c2

%C2f

,−H3∂Sp

%C2f

),

l7 =(− 1

2, 0, 0,

c2

2%Cf,µ0H2

2%Cf,µ0H3

2%Cf,∂Sp

2%Cf

).

(6.5)

Since

∇Φλ1 =(

1, 0, 0,(γ − 1)%c2 − µ0(H2

2 +H23 )

2%2Cf,µ0H2

Cf%,µ0H3

Cf%,c2

2Cf

),

∇Φλ2,3,4,5,6 = (1, 0, 0, 0, 0, 0, 0),

∇Φλ7 =(

1, 0, 0,−(γ − 1)%c2 − µ0(H22 +H2

3 )

2%2Cf,−µ0H2

Cf%,−µ0H3

Cf%,− c2

2Cf

),

(6.6)

by (2.26) we have ciim = 0 for i,m = 2, 3, 4, 5, 6. Via a direct calculation, it holds that

c111(Φ) = −

(12 + γc2

2C2f

+µ0(H2

2+H23 )

%C2f

)< 0. Thus, the first characteristic is genuinely nonlinear.

For other coefficients, noting that li, ri,∇Φri are all regular, one can easily verify that all thecoefficients γiim, γikm are of O(1). The structure of the decomposed equations for H1 = 0 thencoincides with (1.6)-(1.7). Since λ2 = λ3 = λ4 = λ5 = λ6, the corresponding characteristic stripsR2,R3,R4,R5,R6 are also identical. We consider characteristics that propagate in and out ofthe following three characteristic strips: R1,R2,R7. These three strips will be completely

separated when t > t(η)0 . See the following picture.

2η−2η t = 0

t = t(η)0

R1R2R7

Figure 6. Separation of three characteristic strips.


The initial data are given in (5.1)-(5.4) and satisfy the following conditions

W(η)0 = sup

z1(w

(η)1 (z1, 0)) = O(θ), max

i=2,3,4,5,6,7supzi|w(η)i (zi, 0)| ≤ Cη(W

(η)0 )2. (6.7)

The main quantities to be estimated are the same as those in (4.6)-(4.10) except slight changesin the following two quantities:

V2(t) := max sup(x′,t′)/∈R2,

0≤t′≤t

|w2(x′, t′)|, |w3(x′, t′)|, |w4(x′, t′)|, |w5(x′, t′)|, |w6(x′, t′)|,

W (t) := sup(x′,t′)∈R2,

0≤t′≤t

w2, w3, w4, w5, w6.

In the same fashion as in Section 4, for t ∈ [0, T ∗η ) with T ∗η ∼ 1

W(η)0

, we obtain the following

estimates:

S(t) =O(1), J(t) = O(W(η)0 ), V (t) = O

(η(W

(η)0 )2

), W (t) = O

(η(W

(η)0 )2

).

Finally, following the same argument as in Section 5, one can deduce that the H2-norm of thesolution to (6.1) blows up. In conclusion, Theorem 1.2-1.3 also hold when H1 = 0.

7. Proof of H2 ill-posedness for 3D compressible Euler equations (Theorem 1.4)

In this section, we study the case with the absence of magnetic field, i.e., H = 0. The MHDsystem (1.1) then reduces to the 3D compressible Euler equations (1.14)

∂t%+∇ · (%u) = 0,

%∂t + (u · ∇)u+∇p = 0,

∂tS + (u · ∇)S = 0.

To establish ill-posedness via Lindblad-type approach, we study the above system under planarsymmetry

%∂tu1 + %u1∂xu1 + c2∂x%+ ∂Sp∂xS = 0,

%∂tu2 + %u1∂xu2 = 0,

%∂tu3 + %u1∂xu3 = 0,

∂t%+ u1∂x%+ %∂xu1 = 0,

∂tS + u1∂xS = 0.

(7.1)

Recall the definition of vorticity ω := ∇ × u. Under plane symmetry, the vorticity reads ω =(0,−∂xu3, ∂xu2). The system (7.1) therefore allows non-trivial vorticity and entropy.

In a similar fashion, we can also construct examples of shock formation and H2 ill-posednessfor (7.1). In particular, according to the low-regularity local well-posedness result in [46] byWang (See also [13] by Disconzi-Luo-Mazzone-Speck and [52] by Zhang-Andersson), we will seethat the H2 ill-posedness is sharp.

To see the connection, we first review studies on the quasilinear wave equation. The sharplocal well-posedness results were obtained by Tataru-Smith [42] and Wang [45]. And the sharpill-posedness, as mentioned before, was constructed by Lindblad [31]. For Einstein vacuumequations, the sharp local well-posedness was established by Klainerman-Rodnianski-Szeftel [26].


For the 3D compressible Euler equations, the classical local well-posedness result holds ifthe initial data for (u, %) are in Hs with s > 5

2 . In [46], Wang proved the low-regularitylocal well-posedness requiring the initial data of velocity u, density % and vorticity ω satisfying(u, %, ω) ∈ Hs × Hs × Hs′ with 2 < s′ < s. Interested readers are also referred to Disconzi-Luo-Mazzone-Speck [13] and Zhang-Andersson [52]. In [46], it is also conjectured that the sharp

local well-posedness results would be for initial data satisfying (u, %, ω) ∈ Hs×Hs×Hs− 12 with

s > 2. For the 3D incompressible case, the corresponding sharp low-regularity ill-posedness was

proved by Bourgain-Li [4] for initial datum satisfying (u, ω) ∈ H52 ×H

32 and the ill-posedness

is driven by the evolution of ω. Considering the 3D incompressible case as a special scenariofor the 3D compressible Euler equations, Bourgain-Li provided a sharp result with respect tothe regularity of vorticity. Our Theorem 1.4 provides a sharp companion with respect to theregularity of the fluid velocity u and density %. In particular, our initial data leading to the

ill-posedness satisfy (u, %) ∈ H2 ×H2 and ω ∈ C∞ ⊆ H32 . See Figure 7.

s: regularity for the velocity

s′: regularity for the vorticity

2 52

32

2

LWP proved by Wang [46]

(See also Disconzi-Luo-Mazzone

-Speck [13], Zhang-Andersson [52])

LWP by the Conjecture in [46]

Bourgain-Li’s ill-posedness [4]

An-Chen-Yin’s

ill-posedness

Figure 7. Low regularity local well-posedness (LWP) and ill-posedness for3D Euler equations.

With H = 0, denoting Φ := (u1, u2, u3, % − 1, S)T (x, t), the 3D compressible Euler system(7.1) reads

∂tΦ +A(Φ)∂xΦ = 0 (7.2)

with

A(Φ) =

u1 0 0 c2/% ∂Sp/%0 u1 0 0 00 0 u1 0 0% 0 0 u1 00 0 0 0 u1

. (7.3)


λ1 = u1 + c, λ2 = λ3 = λ4 = u1, λ5 = u1 − c (7.4)


with c =√∂%p being the sound speed. And we have a triple eigenvalue λ2 = λ3 = λ4 = u1. The

five linearly independent right eigenvectors are

r1 =

−100−%/c

0

, r2 =

01000

, r3 =

00100

, r4 =

000

−∂Sp/c2

1

, r5 =

−100%/c0

. (7.5)

The dual left eigenvectors are

l1 = (−1

2, 0, 0,− c

2%,−∂Sp

2%c), l2 = (0, 1, 0, 0, 0), l3 = (0, 0, 1, 0, 0),

l4 = (0, 0, 0, 0, 1), l5 = (−1

2, 0, 0,

c

2%,∂Sp

2%c).

(7.6)

With these setup, one can then prove the shock formation and the H2 ill-posedness as in Section6. In particular, with the initial datum of w1 taken as in (5.2), the H1-norms of the velocity u1

and density % will blow up at the shock formation time.Moreover, the vorticity satisfies

∂xu2 =5∑

k=1

wkr2k = w2, ∂xu3 =

5∑k=1

wkr3k = w3.

Therefore, with the following initial data for wi (similar to (5.4)):

w(η)i (z, 0) = θ2

∫Rζ η

10(y)χ(z − y)dy, with i = 2, 3, 4, 5,

we have that the initial datum of ω is smooth and hence belongs to H32 (R3).

Appendix A. Proof of Proposition 3.1

In Proposition 3.1, we analyze and bound the behaviours of ciim, γ44m and γ4

km. In thisappendix, we complete the proof of Proposition 3.1 by proving that the coefficients γiim, γikmwith i 6= 4 satisfy (3.8) when H1 6= 0. This is achieved via a thorough calculation of allthe coefficients. As one will see, the formula of certain coefficients can be very complicated.In spite of these complexities, we show that all these coefficients are of O(1). This is owing to

the careful design of right eigenvectors in (2.9) which eliminates all potentially singular terms,

e.g. (C2a − C2

s )−1, H2

−1, H3−1.

We begin with calculating the coefficients γ1km as below.

γ112 =(λ1 − λ2)

−Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]− C2

a − 3C2s

2Cf (C2f − C2

s )(H3∂H2

Cf −H2∂H3Cf )

,

γ113 =− (λ1 − λ3)

Cf

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)+ ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]


−C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −

2C2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +H3

C2f

∂H3Cf −(

1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs +

(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs

],

γ114 =− (λ1 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

(ρ

γ∂ρC2a

C2f

+2C2

a

C3f

∂SCf

)+

C2a − C2

s

2(C2f − C2

s )

(ρ

γCf∂ρCf −

∂SCfCf

+1

γ

),

γ115 =− (λ1 − λ5)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)+ ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −2C2

aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[Cf

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)+

1

Cf− H2

C2f

∂H2Cf −

H3

C2f

∂H3

− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs +

(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs

],

γ116 =(λ1 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

C2a − 3C2

s

2Cf (C2f − C2

s )(H3∂H2Cf −H2∂H3Cf )

,

γ117 =− (λ1 − λ7)(C2a − C2

s )

C2f − C2

s

[ρ(C2

a − C2f )

C3f

∂ρCf +C2a

C2f

− H2

Cf∂H2Cf −

H3

Cf∂H3Cf

],

γ123 =0, γ124 = 0, γ125 = 0, γ126 =(λ2 − λ6)Cf (Cf + Ca)(C2

a − C2s )

2C2a(C2

f − C2s )

, γ127 = 0,

γ132 =− (λ3 − λ2)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]−

C2f

Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

+Cf (C2

a − C2s )

2C2s (C2

f − C2s )

(H3∂H2Cs −H2∂H3Cs)

,

γ134 =− (λ3 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cs− Cs

)− ∂S

(C2a

Cs− Cs

)]+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )

(ρ∂ρCs − ∂SCs +

Csγ

)+

C2fC

2s

2C2a(C2

f − C2s )

[2C2

a

C3s

∂SCs +1

γ

(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)],


γ135 =− (λ3 − λ5)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2

a

C2s

− 1))2

+2ρC2

aH2

C3s

∂H2Cs +2ρC2

aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[ρ∂ρCsC2s

(C2a

C2s

− 1

)+C2a

C3s

− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ136 =(λ3 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]−

C2f

Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

+Cf (C2

a − C2s )

2C2s (C2

f − C2s )

(H3∂H2Cs −H2∂H3

Cs)

,

γ137 =− (λ3 − λ7)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +

2C2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]−

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ142 =0, γ143 = −(λ4 − λ3)C2

fC2s

2γC2a(C2

f − C2s )

(C2a

C2s

− 1

), γ145 = −

(λ4 − λ3)C2fC

2s

2γC2a(C2

f − C2s )

(C2a

C2s

− 1

), γ146 = 0,

γ147 =(λ4 − λ7)C2

fC2s

2γC2a(C2

f − C2s )

(C2a

C2f

− 1

),

γ152 =− (λ5 − λ2)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

C2f

Cs(C2f − C2

s )(H3∂H2Cs −H2∂H3Cs)

+Cf (C2

a − C2s )

2C2s (C2

f − C2s )


Cs)

,

γ153 =− (λ5 − λ3)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2

a

C2s

− 1))2

+2ρC2

aH2

C3s

∂H2Cs +

2ρC2aH3

C3s

∂H3Cs

]− Cf (C2

a − C2s )

2(C2f − C2

s )

[ρ∂ρCsC2s

(C2a

C2s

− 1

)+C2a

C3s

− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ154 =− (λ5 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cs− Cs

)− ∂S

(C2a

Cs− Cs

)]+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )


Csγ

)−

C2fC

2s

2C2a(C2

f − C2s )

[2C2

a

C3s

∂SCs +1

γ

(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)],

γ156 =(λ5 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

C2f

Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

+Cf (C2

a − C2s )

2C2s (C2

f − C2s )


Cs)

,


γ157 =− (λ5 − λ7)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +

2C2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −H3

C2s

∂H3Cs

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ162 =(λ6 − λ2)Cf (Cf − Ca)(C2

a − Cs)2C2

a(C2f − C2

s ), γ163 = 0, γ164 = 0, γ165 = 0, γ167 = 0,

γ172 =− (λ7 − λ2)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

C2a + C2

s

2Cf (C2f − C2

s )(H3∂H2

Cf −H2∂H3Cf )

,

γ173 =− (λ7 − λ3)

Cf

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +2C2

aH3

C3f

∂H3Cf

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ174 =− (λ7 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]−

C2fC

2s

2C2a(C2

f − C2s )

[1

γ∂ρρ

(C2a

C2f

− 1

)+

2C2a

C3f

∂SCf

]+

C2a − C2

s

2(C2f − C2

s )

(ρ

γCf∂ρCf −

∂SCfCf

+1

γ

),

γ175 =(λ7 − λ5)

Cf

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ176 =(λ7 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

C2a + C2

s

2Cf (C2f − C2

s )(H3∂H2

Cf −H2∂H3Cf )

.

As one can see, the coefficients γ1km have very complex forms. Nevertheless, thanks to the proper

right eigenvectors, there is no singular factors like (C2a − C2

s )−1, H2

−1, H3−1. Hence, every term

in the formula is a simple function of Φ (or its derivatives), which is clearly of O(1). Moreover,the potentially singular factor 1

H22+H2

3in the left eigenvectors, as we explained in Proposition 3.1,

can always be cancelled by a numerator which is either H22 + H2

3 or 0. Then, the combinationof these O(1) terms is also regular.

For i = 2, we obtain

γ221 =λ2 − λ1

4

(CaCf− 1

)2

, γ223 =(λ2 − λ3)

4

(CaCs− 1

)2

, γ224 = −λ2 − λ44γ

,


γ225 =− (λ2 − λ5)

4

(CaCs

+ 1

)2

, γ226 = 0, γ227 = − (λ2 − λ7)

4

(CaCf

+ 1

)2

,

γ213 =0, γ214 = 0, γ215 = 0, γ216 =(λ1 − λ6)(Ca + Cf )

2Cf, γ217 = 0,

γ231 =0, γ234 = 0, γ235 = 0, γ236 = − (λ3 − λ6)(Ca + Cs)

2Cs, γ237 = 0,

γ241 =0, γ243 = 0, γ245 = 0, γ246 = 0, γ247 = 0,

γ251 =0, γ253 = 0, γ254 = 0, γ256 = − (λ5 − λ6)(Ca − Cs)2Cs

, γ257 = 0,

γ261 =λ6 − λ1

4

(C2a

C2f

+ 1

), γ263 = − (λ6 − λ3)(C2

a − C2s )

4C2s

, γ264 = −λ6 − λ44γ

,

γ265 =(λ6 − λ5)(C2

a − C2s )

4C2s

, γ267 = −λ6 − λ74

(C2a

C2f

+ 1

),

γ271 =0, γ273 = 0, γ274 = 0, γ275 = 0, γ276 =(λ7 − λ6)(Cf − Ca)

2Cf.

One can easily see that γ2km are all uniformly bounded.

Next, we calculate γ3km.

γ331 =− (λ3 − λ1)

Cs

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)+ ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −2C2

aH3

C3s

∂H3Cs

]+Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs +

(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs

],

γ332 =(λ3 − λ2)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

3C2f − C2

a

2Cs(C2f − C2


,

γ334 =− (λ3 − λ4)

Cs

2(C2f − C2

s )

[− ρ

γ∂ρ

(C2a

Cs− Cs

)+ ∂S

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

(ρ

γ∂ρC2a

C2s

+2C2

a

C3s

∂SCs

)−

C2a − C2

f

2(C2f − C2

s )

(ρ

γCs∂ρCs −

∂SCsCs

+1

γ

),

γ335 =(λ3 − λ5)(C2

a − C2f )

C2f − C2

s

[ρ(C2

a − C2s )

C3s

∂ρCs +C2a

C2s

− H2

Cs∂H2

Cs −H3

Cs∂H3

Cs

],

γ336 =− (λ3 − λ6)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

3C2f − C2

a

2Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

,


γ337 =− (λ3 − λ7)

Cs

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)+H2∂H2

(C2a

Cs− Cs

)+H3∂H3

(C2a

Cs− Cs

)− ρ(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +2C2

aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −

2C2aH3

C3s

∂H3Cs

]+Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

+H2

C2s

∂H2Cs +H3

C2s

∂H3Cs −(

1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)− 1

Cs

],

γ312 =− (λ1 − λ2)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

[C2s

(C2f − C2

s )Cf−

Cs(C2a − C2

f )

2C2f (C2

f − C2s )

](H3∂H2

Cs −H2∂H3Cs)

,

γ314 =(λ1 − λ4)

Cs

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[ρ

γ∂ρC2a

C2f

+2C2

a

C3f

∂SCf +1

γ

(C2a

C2f

− 1

)]+Cs(C

2a − C2

s )

2(C2f − C2

s )

(ρ

γC2f

∂ρCf −∂SCfC2f

+1

γCf

),

γ315 =− (λ1 − λ5)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

]− Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ316 =(λ1 − λ6)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

[C2s

(C2f − C2

s )Cf−

Cs(C2a − C2

f )

2C2f (C2

f − C2s )

](H3∂H2

Cs −H2∂H3Cs)

,

γ317 =− (λ1 − λ7)

−Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]+

C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2

a

C2f

− 1))2

+2ρC2

aH2

C3f

∂H2Cf +2ρC2

aH3

C3f

∂H3Cf

]−Cs(C

2a − C2

f )

2(C2f − C2

s )

[ρ∂ρCfC2f

(C2a

C2f

− 1

)+C2a

C3f

− H2

C2f

∂H2Cf −

H3

C2f

∂H3Cf

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ321 =0, γ324 = 0, γ325 = 0, γ326 = −Cs(λ2 − λ6)(Ca + Cs)(C

2a − C2

f )

2C2a(C2

f − C2s )

, γ327 = 0,


γ341 =(λ4 − λ1)(C2

a − C2f )C2

s

2γC2a(C2

f − C2s )

, γ342 = 0, γ345 = −(λ4 − λ5)C2

f (C2a − C2

s )

2γC2a(C2

f − C2s )

, γ346 = 0,

γ347 =−(λ4 − λ7)(C2

a − C2f )C2

s

2γC2a(C2

f − C2s )

,

γ351 =− (λ5 − λ1)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +2C2

aH3

C3s

∂H3Cs

]+Cs(C

2a − C2

s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ352 =(λ5 − λ2)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]−

C2a + C2

f

2Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

,

γ354 =(λ5 − λ4)

−Cs

2(C2f − C2

s )

[− ρ

γ∂ρ

(C2a

Cs− Cs

)+ ∂S

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[1

γ∂ρρ

(C2a

C2s

− 1

)+

2C2a

C3s

∂SCs

]+

C2a − C2

f

2(C2f − C2

s )

(ρ

γCs∂ρCs −

∂SCsCs

+1

γ

),

γ356 =(λ5 − λ6)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

C2a + C2

f

2Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

,

γ357 =(λ5 − λ7)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +

2C2aH3

C3s

∂H3Cs

]+Cs(C

2a − C2

s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −H3

C2s

∂H3Cs

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ361 =0, γ362 = −Cs(λ6 − λ2)(Cs − Ca)(C2

a − C2f )

2C2a(C2

f − C2s )

, γ364 = 0, γ365 = 0, γ367 = 0,

γ371 =− (λ7 − λ1)

−Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]+

C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2

a

C2f

− 1))2

+2ρC2

aH2

C3f

∂H2Cf +2ρC2

aH3

C3f

∂H3Cf

]+Cs(C

2a − C2

f )

2(C2f − C2

s )

[ρ∂ρCfC2f

(C2a

C2f

− 1

)+C2a

C3f

− H2

C2f

∂H2Cf −

H3

C2f

∂H3Cf

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ372 =(λ7 − λ2)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

[C2s

(C2f − C2

s )Cf+

Cs(C2a − C2

f )

2C2f (C2

f − C2s )

](H3∂H2

Cs −H2∂H3Cs)

,


γ374 =(λ7 − λ4)

Cs

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]−

C2fC

2s

2C2a(C2

f − C2s )

[ρ

γ∂ρC2a

C2f

+2C2

a

C3f

∂SCf +1

γ

(C2a

C2f

− 1

)]+Cs(C

2a − C2

s )

2(C2f − C2

s )

(ρ

γC2f


+1

γCf

),

γ375 =− (λ7 − λ5)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

]− Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]+

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ376 =(λ7 − λ6)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]−[

C2s

(C2f − C2

s )Cf+

Cs(C2a − C2

f )

2C2f (C2

f − C2s )

](H3∂H2Cs −H2∂H3Cs)

.

Note that the coefficients γ3km take similar forms to that in γ1

km. In the same fashion, one canverify that γ3

km are all regular.Next, for i > 4, the remaining coefficients can be estimated in a similar manner based on

the above calculations. This can be verified easily by noticing the following two facts: first, thepotentially singular 1

H22+H2

3term appears only in the 2nd, 3rd, 5th, 6th components of li and l1,4,7i

are all regular. Since all ∇Φrk · rm are of order 1, it holds that lji · (∇Φrk · rm)j = O(1) withj = 1, 4, 7. Then, we only need to check∑

j=2,3

lji · (∇Φrk · rm)j ,∑j=5,6

lji · (∇Φrk · rm)j .

Second, the components of li satisfy the following symmetry properties

l1,2,3i = l1,2,38−i , l4,5,6,7i = −l4,5,6,78−i .

Therefore, the rest coefficients γikm with i > 4 can be regarded as counterparts of the knownones and thus can be easily verified to be of O(1). Now we also list all the other coefficientswith for i > 4:

γ551 =− (λ5 − λ1)

−Cs

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)+H2∂H2

(C2a

Cs− Cs

)+H3∂H3

(C2a

Cs− Cs

)− ρ(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −

2C2aH3

C3s

∂H3Cs

]− Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +H3

C2f

∂H3Cf −(

1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf


+H2

C2s

∂H2Cs +

H3

C2s

∂H3Cs −

(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)− 1

Cs

],

γ552 =(λ5 − λ2)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

3C2f − C2

a

2Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

,

γ553 =(λ3 − λ5)(C2

a − C2f )

C2f − C2

s

[ρ(C2

a − C2s )

C3s

∂ρCs +C2a

C2s

− H2

Cs∂H2Cs −

H3

Cs∂H3Cs

],

γ554 =− (λ5 − λ4)

Cs

2(C2f − C2

s )

[− ρ

γ∂ρ

(C2a

Cs− Cs

)+ ∂S

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[ρ

γ∂ρC2a

C2s

+2C2

a

C3s

∂SCs

]−

C2a − C2

f

2(C2f − C2

s )

(ρ

γCs∂ρCs −

∂SCsCs

+1

γ

),

γ556 =− (λ5 − λ6)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

3C2f − C2

a

2Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

,

γ557 =(λ5 − λ7)

Cs

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)+ ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −

2C2aH3

C3s

∂H3Cs]

+Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +H3

C2f

∂H3Cf −(

1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs +

(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs

],

γ512 =− (λ1 − λ2)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]−[

C2s

(C2f − C2

s )Cf+

Cs(C2a − C2

f )

2C2f (C2

f − C2s )


,

γ513 =(λ1 − λ3)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

]− Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]+

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ514 =(λ1 − λ4)

Cs

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]−

C2fC

2s

2C2a(C2

f − C2s )

[ρ

γ∂ρC2a

C2f

+2C2

a

C3f

∂SCf +1

γ

(C2a

C2f

− 1

)]+Cs(C

2a − C2

s )

2(C2f − C2

s )

(ρ

γC2f


+1

γCf

),


γ516 =(λ1 − λ6)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]−[

C2s

(C2f − C2

s )Cf+

Cs(C2a − C2

f )

2C2f (C2

f − C2s )

](H3∂H2

Cs −H2∂H3Cs)

,

γ517 =− (λ1 − λ7)

−Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]−

C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2

a

C2f

− 1))2

+2ρC2

aH2

C3f

∂H2Cf +

2ρC2aH3

C3f

∂H3Cf

]−Cs(C

2a − C2

f )

2(C2f − C2

s )

[ρ∂ρCfC2f

(C2a

C2f

− 1

)+C2a

C3f

− H2

C2f

∂H2Cf −

H3

C2f

∂H3Cf

]+

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ521 =0, γ523 = 0, γ524 = 0, γ526 = −Cs(λ2 − λ6)(Ca − Cs)(C2

a − C2f )

2C2a(C2

f − C2s )

, γ527 = 0,

γ531 =− (λ3 − λ1)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +

2C2aH3

C3s

∂H3Cs

]+Cs(C

2a − C2

s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ532 =(λ3 − λ2)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]−

C2a + C2

f

2Cs(C2f − C2


,

γ534 =− (λ3 − λ4)

Cs

2(C2f − C2

s )

[− ρ

γ∂ρ

(C2a

Cs− Cs

)+ ∂S

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[1

γ∂ρρ(

C2a

C2s

− 1) +2C2

a

C3s

∂SCs

]−

C2a − C2

f

2(C2f − C2

s )

(ρ

γCs∂ρCs −

∂SCsCs

+1

γ

),

γ536 =− (λ3 − λ6)

Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]−

C2a + C2

f

2Cs(C2f − C2

s )(H3∂H2

Cs −H2∂H3Cs)

,

γ537 =(λ3 − λ7)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +2C2

aH3

C3s

∂H3Cs

]+Cs(C

2a − C2

s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ541 =−(λ4 − λ1)(C2

a − C2f )C2

s

2γC2a(C2

f − C2s )

, γ542 = 0, γ543 = −(λ4 − λ3)C2

f (C2a − C2

s )

2γC2a(C2

f − C2s )

, γ546 = 0,

γ547 =(λ4 − λ7)(C2

a − C2f )C2

s

2γC2a(C2

f − C2s )

,


γ561 =0, γ562 =Cs(λ6 − λ2)(Cs + Ca)(C2

a − C2f )

2C2a(C2

f − C2s )

, γ563 = 0, γ564 = 0, γ567 = 0,

γ571 =− (λ7 − λ1)

−Cs

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]−

C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2

a

C2f

− 1))2

+2ρC2

aH2

C3f

∂H2Cf +2ρC2

aH3

C3f

∂H3Cf

]+Cs(C

2a − C2

f )

2(C2f − C2

s )

[ρ∂ρCfC2f

(C2a

C2f

− 1

)+C2a

C3f

− H2

C2f

∂H2Cf −

H3

C2f

∂H3Cf

]+

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ572 =− (λ7 − λ2)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

[C2s

(C2f − C2

s )Cf−

Cs(C2a − C2

f )

2C2f (C2

f − C2s )


,

γ573 =(λ7 − λ3)

Cs

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

]− Cs(C

2a − C2

s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]−

C2s (C2

a − C2f )

2C2a(C2

f − C2s )

,

γ574 =(λ7 − λ4)

Cs

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[ρ

γ∂ρC2a

C2f

+2C2

a

C3f

∂SCf +1

γ

(C2a

C2f

− 1

)]+Cs(C

2a − C2

s )

2(C2f − C2

s )

(ρ

γC2f


+1

γCf

),

γ576 =(λ7 − λ6)

−Cs

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

[C2s

(C2f − C2

s )Cf−

Cs(C2a − C2

f )

2C2f (C2

f − C2s )

](H3∂H2

Cs −H2∂H3Cs)

,

γ661 =(λ6 − λ1)

4

(CaCf

+ 1

)2

, γ662 = 0, γ663 =(λ6 − λ3)

4

(CaCs

+ 1

)2

,

γ664 =− λ6 − λ44γ

, γ665 = − (λ6 − λ5)

4

(CaCs− 1

)2

, γ667 = −λ6 − λ74

(CaCf− 1

)2

,

γ612 =− (λ1 − λ2)(Cf − Ca)

2Cf, γ613 = 0, γ614 = 0, γ615 = 0, γ617 = 0,

γ621 =λ2 − λ1

4

(C2a

C2f

+ 1

), γ623 = − (λ2 − λ3)(C2

a − C2s )

4C2s

, γ624 = −λ2 − λ44γ

,

γ625 =(λ2 − λ5)(C2

a − C2s )

4C2s

, γ627 = −λ2 − λ74

(C2a

C2f

+ 1

),

γ631 =0, γ632 =(λ3 − λ2)(Ca − Cs)

2Cs, γ634 = 0, γ635 = 0, γ637 = 0,


γ641 =0, γ642 = 0, γ643 = 0, γ645 = 0, γ647 = 0,

γ651 =0, γ652 =(λ5 − λ2)(Ca + Cs)

2Cs, γ653 = 0, γ654 = 0, γ657 = 0,

γ671 =0, γ672 = − (λ7 − λ2)(Ca + Cf )

2Cf, γ673 = 0, γ674 = 0, γ675 = 0,

γ771 =− (λ1 − λ7)(C2a − C2

s )

C2f − C2

s

[ρ(C2

a − C2f )

C3f

∂ρCf +C2a

C2f

− H2

Cf∂H2

Cf −H3

Cf∂H3

Cf

],

γ772 =− (λ7 − λ2)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

C2a − 3C2

s

2Cf (C2f − C2

s )(H3∂H2Cf −H2∂H3Cf )

,

γ773 =(λ7 − λ3)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cf− Cf

)−H3∂H3

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)+ ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +2C2

aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −

2C2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[Cf (

1

Cf+ρ∂ρCfC2f

)

(C2a

C2s

− 1

)+

1

Cf− H2

C2f

∂H2Cf −

H3

C2f

∂H3

− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs +

(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs

],

γ774 =− (λ7 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

(ρ

γ∂ρC2a

C2f

+2C2

a

C3f

∂SCf

)+

C2a − C2

s

2(C2f − C2

s )

(ρ

γCf∂ρCf −

∂SCfCf

+1

γ

),

γ775 =(λ7 − λ5)

Cf

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)+ ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +2C2

aH3

C3f

∂H3Cf

−(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)− 2C2

aH2

C3s

∂H2Cs −

2C2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

− H2

C2s

∂H2Cs −H3

C2s

∂H3Cs +

(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs

],

γ776 =(λ7 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

C2a − 3C2

s

2Cf (C2f − C2

s )(H3∂H2

Cf −H2∂H3Cf )

,


γ712 =− (λ1 − λ2)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

C2a + C2

s

2Cf (C2f − C2

s )(H3∂H2

Cf −H2∂H3Cf )

,

γ713 =− (λ1 − λ3)

Cf

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +2C2

aH3

C3f

∂H3Cf

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ714 =− (λ1 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cf− Cf

)− ∂S

(C2a

Cf− Cf

)]−

C2fC

2s

2C2a(C2

f − C2s )

[1

γ∂ρρ

(C2a

C2f

− 1

)+

2C2a

C3f

∂SCf

]+

C2a − C2

s

2(C2f − C2

s )

(ρ

γCf∂ρCf −

∂SCfCf

+1

γ

),

γ715 =(λ1 − λ5)

Cf

2(C2f − C2

s )

[H2∂H2

(C2a

Cf− Cf

)+H3∂H3

(C2a

Cf− Cf

)− ρ(C2a

C2s

− 1

)∂ρ

(C2a

Cf− Cf

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2s

− 1

)(C2a

C2f

− 1 + ρ∂ρC2a

C2f

)+

2C2aH2

C3f

∂H2Cf +

2C2aH3

C3f

∂H3Cf

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[H2

C2f

∂H2Cf +

H3

C2f

∂H3Cf −

(1

Cf+ρ∂ρCfC2f

)(C2a

C2s

− 1

)− 1

Cf

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ716 =(λ1 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cf− Cf

)−H2∂H3

(C2a

Cf− Cf

)]+

C2a + C2

s

2Cf (C2f − C2

s )(H3∂H2

Cf −H2∂H3Cf )

,

γ721 =0, γ723 = 0, γ724 = 0, γ725 = 0, γ726 =(λ2 − λ6)Cf (Ca − Cf )(C2

a − C2s )

2C2a(C2

f − C2s )

,

γ731 =(λ3 − λ1)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]−

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +

2C2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]+

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ732 =− (λ3 − λ2)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

C2f

Cs(C2f − C2

s )

(H3∂H2Cs −H2∂H3Cs

)+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )

(H3∂H2

Cs −H2∂H3Cs

),

γ734 =− (λ3 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cs− Cs

)− ∂S

(C2a

Cs− Cs

)]+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )


Csγ

)−

C2fC

2s

2C2a(C2

f − C2s )

[2C2

a

C3s

∂SCs +1

γ

(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)],

γ735 =− (λ3 − λ5)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]


−C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2a

C2s

− 1)

)2

+2ρC2

aH2

C3s

∂H2Cs +

2ρC2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[ρ∂ρCsC2s

(C2a

C2s

− 1

)+C2a

C3s

− H2

C2s

∂H2Cs −H3

C2s

∂H3Cs

]−

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ736 =(λ3 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]+

C2f

Cs(C2f − C2

s )


)+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )

(H3∂H2

Cs −H2∂H3Cs

),

γ741 =(λ4 − λ1)C2

fC2s

2γC2a(C2

f − C2s )

(C2a

C2f

− 1

), γ742 = 0, γ743 =

(λ4 − λ3)C2fC

2s

2γC2a(C2

f − C2s )

(C2a

C2s

− 1

),

γ745 =(λ4 − λ3)C2

fC2s

2γC2a(C2

f − C2s )

(C2a

C2s

− 1

), γ746 = 0,

γ751 =(λ5 − λ1)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2f

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]+

C2fC

2s

2C2a(C2

f − C2s )

[(C2a

C2f

− 1

)(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)+

2C2aH2

C3s

∂H2Cs +

2C2aH3

C3s

∂H3Cs

]+Cf (C2

a − C2s )

2(C2f − C2

s )

[(1

Cs+ρ∂ρCsC2s

)(C2a

C2f

− 1

)+

1

Cs− H2

C2s

∂H2Cs −

H3

C2s

∂H3Cs

]−

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ752 =− (λ5 − λ2)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]−

C2f

Cs(C2f − C2

s )

(H3∂H2

Cs −H2∂H3Cs

)+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )

(H3∂H2

Cs −H2∂H3Cs

),

γ753 =− (λ5 − λ3)

Cf

2(C2f − C2

s )

[ρ

(C2a

C2s

− 1

)∂ρ

(C2a

Cs− Cs

)−H2∂H2

(C2a

Cs− Cs

)−H3∂H3

(C2a

Cs− Cs

)]−

C2fC

2s

2ρC2a(C2

f − C2s )

[1

2∂ρ

(ρ(C2

a

C2s

− 1))2

+2ρC2

aH2

C3s

∂H2Cs +2ρC2

aH3

C3s

∂H3Cs

]− Cf (C2

a − C2s )

2(C2f − C2

s )

[ρ∂ρCsC2s

(C2a

C2s

− 1

)+C2a

C3s

− H2

C2s

∂H2Cs −H3

C2s

∂H3Cs

]−

C2f (C2

a − C2s )

2C2a(C2

f − C2s )

,

γ754 =− (λ5 − λ4)

Cf

2(C2f − C2

s )

[ρ

γ∂ρ

(C2a

Cs− Cs

)− ∂S

(C2a

Cs− Cs

)]+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )


Csγ

)+

C2fC

2s

2C2a(C2

f − C2s )

[2C2

a

C3s

∂SCs +1

γ

(C2a

C2s

− 1 + ρ∂ρC2a

C2s

)],

γ756 =(λ5 − λ6)

Cf

2(C2f − C2

s )

[H3∂H2

(C2a

Cs− Cs

)−H2∂H3

(C2a

Cs− Cs

)]−

C2f

Cs(C2f − C2

s )


)+

Cf (C2a − C2

s )

2C2s (C2

f − C2s )

(H3∂H2

Cs −H2∂H3Cs

),

γ761 =0, γ762 = − (λ6 − λ2)Cf (Cf + Ca)(C2a − C2

s )

2C2a(C2

f − C2s )

, γ763 = 0, γ764 = 0, γ765 = 0.


With all above calculations, we now conclude that all the coefficients ciim and γikm in thedecomposed system (1.4) are of order 1. This finishes the proof of Proposition 3.1.

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Department of Mathematics, National University of Singapore, SingaporeEmail address: [email protected]

Department of Mathematics, National University of Singapore, SingaporeEmail address: [email protected]

Department of Mathematics, Hangzhou Normal University, Hangzhou, ChinaEmail address: [email protected]

arXiv:2110.10647v1 [math.AP] 20 Oct 2021

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