Global existence of strong solution for the compressible - - PowerPoint PPT Presentation

Presentation of the results Idea of the Proof

Global existence of strong solution for the compressible Navier-Stokes equations with large initial data on the irrotational part

Boris Haspot, Universit´ e Paris Dauphine

1 Presentation of the results 2 Idea of the Proof Boris Haspot

Presentation of the results Idea of the Proof

Let us recall the compressible Navier-Stokes equations: Mass equation : ∂tρ + divρu = 0, Momentum equation : ∂t(ρu) + div(ρu ⊗ u) − div(µ(ρ)D(u)) − ∇(λ(ρ)divu) + ∇P(ρ) = 0, Initial data : (ρ, u)/t=0 = (ρ0, u0). Here u = u(t, x) ∈ RN stands for the velocity field, ρ = ρ(t, x) ∈ R+ is the density and D(u) = 1

2 (∇u +t ∇u) the strain tensor.

We denote by λ and µ the two viscosity coefficients of the fluid, which are assumed to satisfy µ > 0 and λ + 2µ > 0 (such a condition ensures ellipticity for the momentum equation and is satisfied in the physical cases where λ + 2µ

N > 0).

In the sequel we shall only consider the shallow-water system which corresponds to: µ(ρ) = µρ with µ > 0 and λ(ρ) = 0. We supplement the problem with initial condition (ρ0, u0).

Boris Haspot

Presentation of the results Idea of the Proof

We aim at solving the system in the case where the data (ρ0, u0) have critical

• regularity. By critical, we mean that we want to solve our system in functional

spaces with invariant norm for the scaling of the equations. It is easy to see that the following transformations: (ρ0(x), u0(x)) − → (ρ0(lx), lu0(lx)), ∀ l ∈ R. (ρ(t, x), u(t, x), P(t, x)) − → (ρ(l2t, lx), lu(l2t, lx), l2P(l2t, lx)). have this property of invariance, provided that the pressure term has been changed in l2P . A good candidate corresponds to the space H

N 2 × (H N 2 −1)N .

Some results of global strong solutions

• A. Matsumura and T. Nishida [80s], Global strong solutions with small initial

data.

• D. Hoff [90s,07], Global weak-strong solution with small initial data with

discontinuous initial density ρ0 ∈ L∞ .

• R. Danchin [2000], Global strong solutions with small initial data in critical

space for the scaling of the equations with u0 ∈ B

N 2 −1

2,1

, (ρ0 − 1) ∈ B

N 2

2,1 .

• Z. Chen et al, F. Charve and R. Danchin, BH [2010,2011], Global strong

solutions with small initial data with u0 ∈ B

N p −1

p,1

, (ρ0 − 1) ∈ B

N p −1

p,1

∩ B

N p

p,1

with 1 ≤ p ≤ N . In particular we have large initial oscillary initial data in LN , u0(x) = φ(x)sin(ε−1x · ω)n, with ω and n some unit vectors and φ ∈ C∞

0 (RN).

Boris Haspot

Presentation of the results Idea of the Proof

How to obtain some global results for a family of large initial data? To do this we are going to consider some irrotational data. More precisely we are interested in constructing irrotational solution. It is natural to deal with such solution, indeed let us mention the case of the so-called compressible Euler system with quantum pressure which is transformed in a non linear Schr¨ dingier equations via the famous transformation of Madelung. More precisely when P(ρ) = 0 we can check that: (ρ1, −µ∇ ln ρ1), is a particular solution of our system (with initial data(ρ0, −µ∇ ln ρ0)) if ρ1 verifies an heat equation:

• ∂tρ1 − µ∆ρ1 = 0,

(ρ1)/t=0 = (ρ1)0. We are going to call (ρ1, −µ∇ ln ρ1) a quasi-solution. The idea in the sequel will consist in working around these profiles. Remark In order to ensure the irrotational form of the solution, we need that the term ∇ ln ρ · Du be a gradient, which implies the structure in ∇ ln ρ for the velocity. Very surprisingly we have some regular effects on the density which is a priori governed by a transport equation.

Boris Haspot

Presentation of the results Idea of the Proof

Littlewood-Paley decomposition corresponds to a dyadic decomposition of the space in Fourier variables. Let (ϕ, ψ) a couple of regular functions such that ϕ has support in C = {ξ ∈ RN/ 3

4 ≤ |ξ| ≤ 8 3 }, and ψ in B(0, 4 3 ) with:

ψ(ξ) +

• l∈N

ϕ(2−lξ) = 1. Let us note h = F−1ϕ, we can define the dyadic blocks by: ∆lu = ϕ(2−lD)u = 2lN

• RN h(2ly)u(x − y)dy if l ∈ Z

Slu =

• k≤l−1

∆ku . We have then modulo the polynoms in S

′(RN):

u =

• k∈Z

∆ku. Definition The Besov space Bs

p,r corresponds to the set of temperated distributions u ∈ S

′

h

such that : uBs

p,r =

l∈Z

2lsr∆lur

Lp

1

r < +∞. Boris Haspot

Presentation of the results Idea of the Proof

Definition Let ρ ∈ [1, +∞], T ∈ [1, +∞] and s1 ∈ R. We define the space Lρ

T (Bs1 p,r) as

follows, u is in Lρ

T (Bs1 p,r) if:

u

Lρ

T (Bs1 p,r) =

l∈Z

2lrs1∆lu(t)r

Lρ

T (Lp)

1

r .

Proposition Let s ∈ R, (p, r) ∈ [1, +∞]2 . Assume that u0 ∈ Bs

p,r and f ∈

L1

T (Bs p,r). Let u

be a solution of:

• ∂tu − µ∆u = f

ut=0 = u0 . Then there exists C > 0 depending only on N, µ, ρ1 and ρ2 such that: u

L1

T (Bs+2 p,r ) ≤ C

• u0Bs

p,r + f

L1

T (Bs p,r)

• .

If in addition r is finite then u belongs to C([0, T], Bs

p,r).

Boris Haspot

Presentation of the results Idea of the Proof

We now search some solutions written under the following form ln ρ = ln ρ1 + h2 (ρ = ρ1eh2 , with ρ1 = 1 + q1 ) and u = −µ∇ ln ρ1 + u2 , we have then modulo that the density does not admit vacuum:      ∂t ln ρ + u · ∇ ln ρ + divu = 0, ∂tu + u · ∇ − µ∆u − µ∇ ln ρ · Du + ∇F(ρ) = 0, (ln ρ, u)/t=0 = (ln ρ0, u0). (1) By using the fact that (ρ1, u1) = (ρ1, −µ∇ ln ρ1) is a quasi solution with:

• ∂tρ1 − µ∆ρ1 = 0.

ρ1(0, ·) = ρ1

0.

we can rewrite the system under the following form:            ∂th2 + u · ∇h2 + divu2 = −u2 · ∇ ln ρ1, ∂tu2 + u · ∇u2 − µ∆u2 + a∇h2 = −a∇ ln ρ1 − u2 · ∇u1 + µ∇ ln ρ1 · Du2 +µ∇h2 · Du1 + µ∇h2 · Du2, (h2, u2)/t=0 = (h2

0, u2 0).

(2) where we have assumed to simplify that P(ρ) = aρ. We are going to solve the system (2) with small initial data (h2

0, u2 0) and with large ρ1 .

Boris Haspot

Presentation of the results Idea of the Proof

Theorem Let ρ0 = (ρ1)0eh2

0 and u0 = −µ∇ ln(ρ1)0 + u2

0 . Moreover we assume that

(ρ1)0 ≥ c > 0, (ρ1 − 1)0 ∈ B

N 2 −2

2,1

∩ B

N 2

2,1 , h2 0 ∈ B

N 2

2,1 and u2 0 ∈ B

N 2 −1

2,1

. Then it exists ε > 0, C > 0 and l > 0 large enough (depending on (ρ1 − 1)0

B

N 2 2,1

such that if:

• k≥l

∆k(ρ1 − 1)0

B

N 2 −2 2,1

≤ ε and h2

B

N 2 2,1

+ u2

B

N 2 −1 2,1

≤ ε, (3) then it exists a global strong solution (ρ, u) of the shallow-water system under the following form: ρ = ρ1eh2 and u = −µ∇ ln ρ1 + u2 with:

• ∂tρ1 − µ∆ρ1 = 0,

(ρ1)t=0 = (ρ1)0. (4) and such that: h2 ∈ C(R+, B

N 2

2,1) ∩ L1(R+, B

N 2

2,1)

and u2 ∈ C(R+; B

N 2 −1

2,1

) ∩ L1(R+, B

N 2 +1

2,1

).

Boris Haspot

Presentation of the results Idea of the Proof

Remark We have a super-critical condition of smallness for the initial density, more precisely we can write this condition under the following form: (ρ1 − 1)0

B

N 2 −2 2,1

≤ Cexp(−C(ρ1 − 1)0

B

N 2 2,1

). We can choose initial data with α and β well-chosen of the form: q1

0(x) = 1

εβ ei x3

ε f(x1, x2

εγ , x3), with (ρ1)0 =

1 εβ + q1 0 and f ∈ S(RN).

In particular we allows initial density with large L∞ norm. Remark In particular we can obtain global strong solution in dimension N = 2 with large initial data in the energy spaces. We also refer to a remarkable work of Kazhikhov and Waigant for other viscosity coefficients. Remark In a very surprising way the density is decomposed in a small part h2 and in a regular part ρ1 . This regularizing effect remains quite surprising.

Boris Haspot

Presentation of the results Idea of the Proof

In the sequel we are going to show the global existence for (h2, u2). First step: Construction of approximate solutions We regularize the initial data: (h2)n

0 = Snh2 0, (u2)n 0 = Snu2 0.

We obtain then a solution (hn

2 , un 2 ) on the interval (0, Tn) such that:

hn

2 ∈

C([0, Tn], BN

2,1 ∩ B

N 2 −1

2,1

) un

2 ∈

C([0, Tn], B

N 2 −1

2,1

) ∩ L1([0, Tn], B

N 2 +1

2,1

). Second step: Uniform estimates We are going to obtain uniform estimates on (hn

2 , un 2 ) in order to show that

Tn = +∞. To do this, we need to distinguish the behavior in low and high frequencies. We need to study the following system:      ∂thn + un · ∇hn + divun

2 = F n 1 ,

∂tun

2 − µ∆un 2 + a∇hn = Gn 1 ,

hn

0 = h0, (un 2 )/t=0 = (u2)n 0 .

(5) Let us mention that it is necessary to include in the study the convection term in

• rder to not lose derivative on the density.

Boris Haspot

Presentation of the results Idea of the Proof

Estimate on the linearized system in Besov spaces: Proposition Let s ∈ R. Let (h2, u2) a solution of the previous system. It exists a constant C such that: (h2, u2)(t)(Bs−1

2,1 ∩Bs 2,1)×Bs−1 2,1 +

t (h2, u2)(s)Bs

2,1×Bs+1 2,1 ds

≤ C

• (h2

0, u2 0)(Bs−1

2,1 ∩Bs 2,1)×Bs−1 2,1 +

t e−V (s)(F1, G1)(s)(Bs−1

2,1 ∩Bs 2,1)×Bs−1 2,1 ds

• .

with V (T) = T

0 ∇u(s)L∞ds.

To prove this proposition we need to distinguish the behavior in low frequencies and in high frequencies. In high frequencies, we introduce a effective velocity v2 which diagonalize the system: v2 = u2 − a(∆)−1∇h2. We can rewrite the system as follows:

• ∂th2 + u · ∇h2 + ah2 + divv2 = F1,

∂tv2 − µ∆v2 = G1 + f, (6) with f a low regularity order term which can be absorbed in high frequencies.

Boris Haspot

Presentation of the results Idea of the Proof

It means that roughly speaking we have only to deal with a heat equation on v2 and a transport equation with a damping term for h2 . In low frequencies we are directly working with the unknowns (h2, u2) and we are going to take advantage of the regularizing effect on the density h2 (in low frequencies). In particular it allows us to consider the convection term u · ∇h2 as a remainder term, we have to study then the following system:

• ∂th2 + divu2 = F1 − u · ∇h2,

∂tu2 − µ∆u2 + a∇h2 = G1, (7) It remains to study the Green kernel associated to this linear system by using the Fourier transform. It concludes the proof of the proposition 2.1. We are going to use this proposition in order to obtain uniforms bound estimates

• n (hn

2 , un 2 ). Let us define the following norm:

E(h, u) = h

• L∞(B

N 2 −1 2,1

∩B

N 2 2,1)∩

L1(B

N 2 2,1)

+ u

• L∞(B

N 2 −1 2,1

)

+ u

• L1(B

N 2 +1 2,1

)

, When we apply the proposition 2.1 to our system and by using paraproduct laws for the Besov spaces in order to treat the remainder term, we obtain the following inequality: E(hn

2 , un 2 ) ≤ Ce(ρ1,−µ∇ ln ρ1+un

2 )E

h2

B

N 2 −1 2,1

∩B

N 2 2,1

+ u2

B

N 2 −1 2,1

+a∇ ln ρ1

• L1(B

N 2 −1 2,1

)

+ E(hn

2 , un 2 )

• .

Boris Haspot

Presentation of the results Idea of the Proof

At this level we can apply a classical bootstrap argument in order to obtain global estimate for (hn

2 , un 2 ).

Let us observe that to study the system under his eulerian form we need to control the vacuum, this is done because ρ1 verifies an heat equation, we can apply a maximum principle in order to prove that ρ1 ≥ c > 0. Let us point out now that here as ρ1 verifies an heat equation and that we control the vacuum on ρ1 , we have: ∇ ln ρ1

• L1(B

N 2 −1 2,1

)

≤ C(ρ0)1

B

N 2 −2 2,1

. We observe then that we have a smallness condition on ρ1 which is supercritical. More precisely we have a condition of the form: (ρ1 − 1)0

B

N 2 −2 2,1

≤ Cexp(−C(ρ1 − 1)0

B

N 2 2,1

). We can choose initial data with α and β well-chosen of the form: q1

0(x) = 1

εβ ei x3

ε f(x1, x2

εγ , x3), with (ρ1)0 =

1 εβ + q1 0 and f ∈ S(RN). In particular we allows initial density with

large L∞ norm.

Boris Haspot

Presentation of the results Idea of the Proof

Third step: Existence of solutions Finally by using compactness arguments involving the Ascoli theorem, we show that (hn

2 , un 2 ) converges weakly in E to (h2, u2) a solution.

Fourth step: Uniqueness The uniqueness is classical (we refer to Danchin). THANK YOU FOR YOUR ATTENTION!

Boris Haspot