Mathematical Issues in Navier-Stokes Equations

Hi Jun Choe

CMAC(Center for Mathematical Analysis and Computations)Yonsei University, Seoul, Korea

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Abstract.The macro motion of fluid is governed by compressible orincompressible Navier-Stokes equations. The Mach numberscale decides the compressible or incompressible nature. In thistalk, our main concerns are mathematical issues ofNavier-Stokes equations like existence and singularity. Weintroduce the existence theories of weak or strong solutions forcompressible flows and discuss current trends including finiteblow up, too. For the incompressible flows, we discuss possiblesingularities and our efforts to answer the millennium problem.

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Contents.

• 1, Introduction and Mathematical Models

• 2. Compressible Navier-Stokes equations

• 3. Incompressible Navier-Stokes equations

• 4. Various Questions

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1. Introduction and Mathematical Models

Fluid is something like water, air and oil that change theirshapes according to container. Because all materials are madeof particles like atoms, the motion of fluid is modeled fromHamiltonian dynamics, statistical mechanics and continuummechanics.In this talk we address mathematical foundation, mechanicalequations and challenges. Although most mathematicalquestions are old and intensely studied, some of them standfirmly open. We introduce related mathematical progressesand trend.

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Compressible Navier-Stokes Equations

Set ρ density, p pressure, u velocity, and θ temperature. Wehave heat conducting compressible Navier-Stokes equations;

∂tρ+∇ · (ρu) = 0,

∂t(ρu) +∇ · (ρu⊗ u) +∇p(ρ, θ)= µ∆u+ (ν + µ)∇div(u)

∂t

(ρ(

1

2|u|2 +

3

2θ)

)+∇ ·

((ρ(

1

2|u|2 +

5

2θ)u

)= κ∆θ + λ∇ · (D(u) · u),

where µ and ν are viscosity coefficients satisfying

µ > 0, 3ν + 2µ ≥ 0,

κ is heat conductivity and λ is energy dissipation coefficient.5 / 30

When the temperature θ is constant, we have isentropiccompressible Navier-Stokes equations;

∂tρ+∇ · (ρu) = 0,

∂t(ρu) +∇ · (ρu⊗ u) +∇p= µ∆u+ (ν + µ)∇div(u)

p = p(ρ)

µ > 0, 3ν + 2µ ≥ 0,

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Incompressible Navier-Stokes Equations

Further, when density ρ is constant, we have incompressibleNavier-Stokes equations;

ut − ν∆u+ (u · ∇)u+∇p = 0

div(u) = 0.

If the viscosity is zero, we obtain incompressible Eulerequations;

ut + (u · ∇)u+∇p = 0, div(u) = 0.

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2. Compressible Navier-Stokes Equations

In case of isentropic flow following gas law

p = aργ, γ ≥ 1.8

P.L.Lions initiated the global existence of weak solution withappropriate initial condition;

ρ ∈ L∞(0, T ;Lγ(R3)), u ∈ L2(0, T ;W 1,20 (R3)∫

1

2ρ|u|2+

a

γ − 1ργdx+

∫ T

0

∫µ|∇u|2+(µ+ν)|div(u)|2dxdt ≤ 0.

• P.-L. Lions, Mathematical topics in fluid dynamics, Vol. 2,Compressible models, Oxford Science Publication, Oxford,1998.

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We have been trying to reduce the gas constant γ to 1. In thisefforts, Cech group reduced γ by

γ > 1.5.

• Eduard Feireisl, Antonin Novotny and Hana Petzeltova, Onthe Existence of Globally Defined Weak Solutions to theNavier-Stokes Equations, J. math. fluid mech. 3 (2001)358-392.

Recently there is a claim for the case γ = 1.5.

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For the heat conducting compressible Navier-Stokes equations,French Bresch and Desjardins proved glabal existence of weaksolution under the assumption that ν and µ are function of ρunder certain growth, heat conductivity κ is a function of ρand θ with appropriate condition and pressure is idealpolytropic gas type.

• Bresch, Didier; Desjardins, Benot, On the existence of globalweak solutions to the Navier-Stokes equations for viscouscompressible and heat conducting fluids. J. Math. Pures Appl.(9) 87 (2007), no. 1, 5790.

Still it is unknown of global existence of weak solutions forconstant parameter cases.

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Before global existence of weak solution was known, there hadbeen a lot of works for short time existence of strongsolutions. We briefly review some. Local existence of a strongsolution and global existence with the initial data close to anequilibrium state have been well developed.• Danchin, R.: Local theory in critical spaces for compressibleviscous and heatconductive gases. Comm. Partial Differ. Equ.26, 11831233 (2001)•Matsumura, A., Nishida, T.: The initial boundary valueproblems for the equations of motion of compressible andheat-conductive fluids. Comm. Math. Phys. 89, 445464(1983)

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A remarkable fact is that a strong solution always blow upwhen the initial data has compact support.• Xin, Z.: Blowup of smooth solutions to the compressibleNavierStokes equation with compact density. Comm. PureAppl. Math. 51, 229240 (1998)

Nonetheless Korean group proved local existence of strongsolutions when there is vacuum;• Cho, Yonggeun; Choe, Hi Jun; Kim, Hyunseok, Uniquesolvability of the initial boundary value problems forcompressible viscous fluids. J. Math. Pures Appl. (9) 83(2004), no. 2, 243275.

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Theorem

Let Ω be a bounded smooth domain or R3 and q ∈ (3, 6],Suppose ρ0 is nonnegative and belongs toW 1,q(Ω)∩H1(Ω)∩L1(Ω). Suppose u0 ∈ D1

0(Ω)∩D2(Ω) andsatisfies the compatibility condition

Lu0 +∇p(ρ0) =√ρ0g

for a vector field g ∈ L2(Ω). Then there exist a timeT ∈ (0,∞] and unique solution

(ρ, u) ∈ C([0, T ) : H1∩W 1,q(Ω))×C([0, T ) : D2(ω)∩L2([0, T ) : D2,q(Ω)).

Moreover, if we let T ∗ be maximal existence time of solutionand T ∗ is finite, then there holds

lim supt→T ∗

||ρ||W 1,q(t) + ||u||D1(Ω) =∞.13 / 30

This result initiated recent activity ofFinite Time Blow UP,Regularity of Weak Solution,Propagation of Singularity.

• Hoff,D.: Discontinuous solutions of the NavierStokesequations for multidimensional flows of heat-conducting fluids.Arch. Rational Mech. Anal. 139, 303354 (1997)• Hoff, D.: Global solutions of the NavierStokes equations formultidimensional compressible flow with discontinuous initialdata. J. Differ. Equ. 120, 215254 (1995)

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After our result, there have been quite a few works for finitetime blow up by many Chinese researchers including Z. Xin.We like to mention a result by Sun-Wang-Zhang(2011); Underassumption ν < 7µ, the strong solution blows up at t = T ifand only if

limt→T||ρ||L∞ =∞.

There have been activities to remove unphysical assumptionν < 7µ.

As a final remark, we like to mention the weak-stronguniqueness.

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3. Incompressible Navier-Stokes Equations

The existence of weak solution due to Leray(1934) might beone of the most significant event in analysis.Weak solution concept was introduced for the first time.• Leray, Jean, Sur le mouvement d’un liquide visqueuxemplissant l’espace. Acta Math. 63 (1934), no. 1, 193 - 248.

Introducing Reynolds number Re, we obtain incompressibleNavier-Stokes equations;

ut −1

Re∆u+ (u · ∇)u+∇p = 0, div(u) = 0

and we consider only Cauchy problem.

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Definition

Suppose u ∈ L2(Ω). For all solenoidal φ ∈ C∞0,σ(Ω), thefollowing equality holds;∫u · φtdxdt+

1

Re

∫∇u : ∇φdxdt+

∫u⊗ u : ∇φdxdt = 0.

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Theorem

There is a weak solution u

u ∈ L∞(0,∞;L2(R3)) ∩ L2(0,∞;W 1,2(R3))

satisfying energy inequality

1

2

∫|u|2dx(T ) +

1

Re

∫ T

0

∫|∇u|2dxdt ≤ 1

2

∫|u|2dx(T )

for all T ≥ 0.

Note we do not know the energy equality holds, yet. If strictinequality holds, it is called turbulent solution and energydissipation might occur in small scale.

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Millennium problem of Clay Institute

”The Navier-Stokes equations is the equation which governsthe flow of fluids such as water and air. However, there is noproof for the most basic questions one can ask: do solutionsexist, and are they unique? Why ask for a proof? Because aproof gives not only certitude, but also understanding.”From strong-weak uniqueness, we try to show regularity ofweak solutions.

Definition

u is a strong solution if

u ∈ L∞(0,∞;H1).

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Due to scaling structure, if (u(x, t), p(x, t)) is a solution, then

(ur(x, t), pr(x, t)) = (ru(rx, r2t), r2p(rx, r2t))

is also solution.

There are many scale invariant spaces;

L∞(0, T ;L3(R3))); L∞(0, T ;H1/2(R3));

Lq(0, T ;Lp(R3)), 1 =3

p+

2

q; · · · ,

namely, ||u||S = ||ur||S if S is scale invariant space.

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Leray(1934) proved that the singular time set

S = τ : limt→τ−

||∇u||L2 =∞

has at most 0.5 Hausdorff dimension.

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• Caffarelli-Kohn-Nirenberg(1982) obtained a regularitycriterion; There is an absolute constant ε such that If

lim supr→0

1

r

∫Qr(x,t)

|∇u|2dxdt < ε,

then u is bounded in a neighborhood of (x, t), where Qr(x, t)is a parabolic cylinder centered at (x, t). So the singular sethas at most 1 parabolic Hausdorff dimension.

• Later, Choe-Lewis(2000) improved by a logarithmic factor.

• Recently, Dr. Yang, Minsuk is working on Minkowskiidimension.

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Mild solution

Definition

u ∈ L∞(0,∞;L2) is a mild solution if

u(x, t) = et∆u0(x) +

∫ t

0

e(t−s)∆Pdiv(u⊗ u)ds

for almost all (x, t), where P (f)i = RiRjfj and Ri is i-thRiesz operator

A remarkable theorem of Kato appeared in 1984.• Kato, Tosio, Strong L p -solutions of the Navier-Stokesequation in R m , with applications to weak solutions. Math.Z. 187 (1984), no. 4, 471480.The theorem says if the initial data u0 is L3, then there is asmall time T > 0 such that a strong solution exists in (0, T ).

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Indeed it is forward regularity for critical space L3. Theexistences and regularity of mild solutions in L∞(0, T ; H1/2)and several Besov space are under investigation;• Kenig, Carlos E.; Koch, Gabriel S. An alternative approachto regularity for the Navier-Stokes equations in critical spaces.Ann. Inst. H. Poincar Anal. Non Linaire 28 (2011), no. 2,159187.•H. Koch, D. Tataru, Well-posedness for the NavierStokesequations, Adv. Math. 157 (2001) 2235.

Note H1/2 ⊂ L3.

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Type I singularity

It turns out that the critical case of Serrin class L∞(0, T ;L3)is regular class.• Iskauriaza, L.; Sergin, G. A.; Shverak, V. L3,∞ -solutions ofNavier-Stokes equations and backward uniqueness. (Russian)Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3–44.

There is no self-similar solution of the type

u(x, t) =1√T − t

U( x√

T − t

)for a smooth function U in L∞(0, T ;L2) ∩ L2(0, T ;H1).• Necas, J.; Ruzicka, M.; Sverak, V. On Leray’s self-similarsolutions of the Navier-Stokes equations. Acta Math. 176(1996), no. 2, 283-294.

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u has Type I singularity at (x0, t0) if

ess supQr(x0,t0)

|x− x0||u(x, t)| <∞

ess supQr(x0,t0)

√|t0 − t||u(x, t)| <∞.

There is no type I singularity in space for axisymmmetric flow.There is no type I singularity in time for axisymmmetric flowwithout swirl.• Seregin, G.; Sverk, V. On type I singularities of the localaxi-symmetric solutions of the Navier-Stokes equations.Comm. Partial Differential Equations 34 (2009), no. 1-3,171-201.

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Note that1

|x|∈ L3

weak but1

|x|/∈ L3.

So to understand type I singularity we study L∞(0,∞;L3weak).

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Theorem

Theorem 2. Suppose u is a weak solution to the Cauchyproblem in L∞(0,∞;L3

weak). Then there exist at most finitenumber N of singular points at any singular time t.

• ON REGULARITY AND SINGULARITY FORL∞(0, T ;L3,w(R3)) SOLUTIONS TO THE NAVIERSTOKESEQUATIONS, HI JUN CHOE, JRG WOLF, MINSUK YANG

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4. Various Questions

• Mach limit• Vanishing viscosity• Hydrodynamic limit• Asymptotics• Turbulence model• Global attractor and inertial manifold

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Thank you for yourattention!

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