On the local well-posedness of compressible viscous flows with bounded density

Transcription

1 On the local well-posedness of compressible viscous flows with bounded density Marius Paicu University of Bordeaux joint work with Raphaël Danchin and Francesco Fanelli Mathflows 2018, Porquerolles September 7, 2018

2 THE BAROTROPIC NAVIER-STOKES EQUATIONS

3 Barotropic Navier-Stokes t ρ + div(ρ u) = 0 t (ρ u) + div ( ρ u u ) + P(ρ) µ u λ div u = 0 (t, x) R + Ω, Ω R n, n 1 Density: ρ(t, x) 0 Velocity field: u(t, x) R n Pressure: P(ρ) R, P (ρ) > 0 Boyle s law: P(ρ) = A ρ γ (γ > 1) Viscosity coefficients: µ > 0 and ν := µ + λ > 0

4 Weak solutions theory Energy inequality: ρ(t) u(t) 2 L + H ( ρ(t), ρ ) 2 L µ t 0 u(τ) 2 L 2 dτ + λ t 0 div u 2 L 2 dτ C ( ρ 0 u 0 2 L 2 + H ( ρ0, ρ ) L 1 ) ρ 0 fixed constant reference state H := internal energy functional H (ρ) = P (ρ)/ρ, H(ρ) = H (ρ) = 0 P > 0 = H ( ρ, ρ ) (ρ ρ) 2

5 P(0) = 0, P (ρ) > 0, P (ρ)/ρ γ 1 ρ + A > 0 Theorem ( P.-L. Lions 1993 ) γ γ(n) > 1 = global in time finite-energy weak solutions (ρ, u) Feireisl & Novotný & Petzeltová 2001: improved the value of γ(n) Global weak solutions for non-monotone pressure laws (a) Feireisl 2001 (b) Ducomet, Feireisl, Petzeltova, Straskraba 2004 (c) Bresch & Jabin 2015

6 Strong solutions theory 1) Nash 1962: 0 < ρ ρ 0 ρ = local existence and uniqueness of classical solutions 2) Solonnikov 1976: local well-posedness in smooth bounded domain 3) Danchin 2001: ρ 0 ρ small or ρ 0 ρ > 0 = local well-posedness in critical spaces Danchin 2014: Lagrangian coordinates Global well-posedness (i) P (ρ) > 0 (ii) ρ 0 ρ and u 0 small Matsumura & Nishida 1980: H 3 regularity Danchin 2000: critical spaces

7 Hoff s solutions for shock-data 1995: global weak solutions in dimensions 2 and 3 (i) ρ 0 ρ L2 L and u 0 L 2 small (ii) u 0 L 4 if N = 2 and u 0 L 8 if N = 3 (iii) P(ρ) = A ρ 2002: in R 2, propagation of density discontinuities for all times (i) P(ρ) = A ρ (ii) K 0 = Jordan curve of class C 1+α (iii) ρ 0 ρ L 2 C and u pw α 0 H β L small q (iv) [ρ 0 ] small = global weak solutions satisfy: a) u LT(L 1 ) and u admits a unique flow ψ b) K(t) := ψ t (K 0 ) is C 1+α c) ρ(t) ρ and div u are Cpw α

8 2006: uniqueness in R n (n = 2, 3) u 1 u 2 L 2 ([0,T] R n ) + sup ρ 1 (t) ρ 2 (t) H 1 (R n ) [0,T] (i) u 1, u 2 L 1 T(L ) Lagrangian coordinates (ii) hypothesis for the pressure: ( ) P(ρ1 ) P(ρ 2 ) P(ρ) = A ρ or ρ 1 ρ 2 L p H 1 control for P(ρ) H 1 control for ρ

9 Remarks (1) finite energy weak solutions, with ρ(t) ρ L (2) additional regularity ( with time weights ) for = u, ω := curl u, u := t u + u u The key quantity is F := ν div u (P(ρ) P(ρ)) the effective viscous flux F(t) has no jump on the interface of the discontinuities. Our goals: t > 0, ω(t), F(t) H 1 (R n ) and F(t) C α (R n ) (1) improve the existence of local weak solutions for bounded densities and general P(ρ) (2) Uniqueness ( general P(ρ), discontinuous densities) (3) Propagation of singularities ( higher dimensions, general P(ρ) )

10 WELL-POSEDNESS FOR BOUNDED DENSITIES

11 The setting t ρ + div(ρ u) = 0 t (ρ u) + div ( ρ u u ) + P(ρ) + Lu = 0 (t, x) R + R n, n 1 Lamé operator: Lu := µ u λ div u Pressure: P(ρ) W 1, loc, P(1) = 0 σ := ρ 1 v := (Id ν ) 1 P(ρ) (ν = µ + λ) Effective velocity (effective viscous flux): w := u v

12 The existence statement Theorem ( Danchin & Fanelli & P ) n < p < +, 1 < r < 2p/(2p n) σ 0 (L p L ) (R n ), σ 0 L ε w 0 Ḃ 2 2/r p,r = (ρ, u) solution on [0, T] R n such that: (i) σ L ([0,T] R n ) 4 ε (ii) σ C ( [0, T]; L q), v C ( [0, T]; W 1,q) (q [p, + [ ) (iii) w C ( [0, T]; Ḃ 2 2/r ) p,r L r 0 T (L ), with w L r 1 T (Lp ) and t w, 2 w L r T (Lp ) Moreover, if σ 0, u 0 L 2, and P (ρ) > 0 the energy E(t) is conserved in time, where E(t) = 1 ρ(t) u(t) H ( ρ(t) ) L 2 L 1+ µ u 2 Lt 2 (L 2 ) + λ div u 2 Lt 2 (L 2 )

13 A maximal regularity approach t σ + div ( (σ + 1) u ) = 0 t u + Lu = (σ + 1) u u σ t u P (!) We cannot use the maximal regularity approach directly on u as we need P L p L q and so σ L p L q ) = effective velocity w t σ + (w + v) σ + (σ + 1) (div w + div v) = 0 t w + Lw = σ t w + (σ + 1) F F : Lv + P, u u, t v With the definition v = ν 1 P, we have F : Lv + P = 0.

14 On the proof t σ + (w + v) σ + (σ + 1) (div w + div v) = 0 t w + Lw = σ t w + (σ + 1) F F : Lv+ P = smooth term in P, (w+v) (w+v), t v t v = (Id ν ) 1 t P = (Id ν ) 1 (g(ρ) div u div ( P u )) with g(ρ) := P(ρ) ρp (ρ). 1) The density: transport estimates 2) w : maximal regularity estimates t Duhamel s formula: w(t) = e Lt w 0 + e L(t τ) f (τ) dτ 0 Initial datum: t s/2 e t z L p z L r (R +;dt/t) Ḃ s p,r External force: continuity on LT(L r p ) of the operators t ( f Id,, 2 ) e (t τ) f (τ) dτ 0

15 L p (L q ) Maximal regularity Recall the maximal regularity for parabolic equation: t e (t s) f (s)ds C f L p L q. L p L q 0 In particular, u 0 B 2 2 r p,r if and only if e t u 0 L r (R + ; L p (R d )). u L L p 1 t (L q 1 x ), u L L p 2 t (L q 2) u L L p 3 t (L q 3 x ), Π L L p 3 t (L q 3 x ) - Let X = {u u L r 1(L p 1), u L r 2(L p 2), u L r 3(L p 3)}. - if u X then f = ( u u + a u + a Π) L p 3 t (L q 3 x ) Applying the maximal regularity estimates, we obtain u X u 0 Ḃ2 2 r p,r + a L u X + C(T) u 2 X.

16 Some remarks (1) z Ḃ 2/r p,r e t z L r( R + ; L p) = slightly sub-critical regularity for w 0 B 2 2 r p,r Critical space for velocity field: Ḃ 1+n/p p,r 2 2/r = 1 + n/p 2/r + n/p = 3 (!) Here, 2/r + n/p < 3 (2) 2 w L r (L p ), p > n = w L r 1 T (L ) For uniqueness, Lagrangian coordinates = we need u L 1 T(L ) u = w + v = w + 2 (Id ) 1 P (!) We do NOT have v = 2 (Id ) 1 P L 1 T(L )

17 Vortex patches problem for 2-D Euler 2-D incompressible Euler vorticity formulation t ω + u ω = 0 u = ( ) 1 ω Theorem ( Yudovich 1963 ) ω 0 L 1 L =! global weak solution ω L T (L1 L ) Vortex patch: ω 0 = 1 D0, D 0 R 2 bounded = ω(t) = 1 D(t), D(t) := ψ t (D 0 )

18 Theorem ( Chemin 1991, 1993 ) ε ]0, 1[, D 0 C 1,ε = then: (i) D t C 1,ε at any time t (ii) u L 1 T (L ) ( via tangential regularity ) Idea: Control of the co-normal regularity In general, if ω L = u is only Log-Lipshitz. If X(t, ) is the tangential vector field to D t, then X ω = 0. Let ω L and X ω C ɛ 1 (X λ C ε ; I(X) = inf x sup λ X λ (x) > 0) i j 1 ω L Using the Biot-Savart laws u L C I(X) ( ω L + Xω C ε 1) C I(X) ω L ln ( e + Xω C ε 1 ω L )

19 Generalizations (a) higher dimensions Gamblin & Saint-Raymond 1995, Danchin 1999 (b) viscous fluids Danchin 1997, Hmidi 2005 (c) density-dependent incompressible Euler Fanelli 2012 (d) non-homogeneous incompressible Navier-Stokes Liao & Zhang 2016, 2018, Danchin & Zhang 2017, Gancedo & García-Juárez 2017, P. & Zhang ( variable viscosity case) 2017,...

20 Uniqueness statement Theorem ( Danchin & Fanelli & P ) Previous hypotheses ( for existence ) X 0 non-degenerate family of vector fields in L,p X0 ρ 0 L p =! (ρ, u) solution on [0, T] R n such that: (i) previous properties hold true (ii) u L 1 T (L ) ψ t = flow (iii) X t := transported of X 0 remains non-degenerate and L,p (iv) Xt ρ(t) L p

21 Sketch of the proof 1) Propagation of tangential regularity for the density Given X X t : t X + u X = X u and [ X ; t + u ] = 0 We have t div (ρ X) + u div (ρ X) = div u div (ρ X). = if u L 1 T (L ), then OK propagation of: L bounds for X L p bounds for X ρ X P(ρ) (!) Propagating L p regularity for X X u L p X u = X w + X 2 (Id ) 1 P(ρ). We already have 2 w L r L p. We need to bound X 2 (Id ) 1 P(ρ) L p X 2 (Id ) 1 P(ρ) L p C(X) ( X L p P(ρ) L + X P(ρ) L p).

22 Lipschitz bounds on the velocity As u = w + 2 (Id ν ) 1 P(ρ), we need to bound 2 (Id ν 1 )P in L. We use the control of the tangential derivative and the fact that 2 (Id ν ) 1 = a(x)id + b(x) X (Id ν ) 1 2 (Id ) 1 P(ρ) L C(X) ( P(ρ) L + X L p X P L p). 2 (Id ) 1 P(ρ) L T0 (L ) C 0, w L 1 t (L ) (t C 1 1/r 1 w r L 1 T (L p 1) + t1 1/r 2 2 w ) r L 2. T (L p ) which implies, the following control, for all fixed t [0, T 0 ]: u L 1 t (L ) (t C 1 1/r 1 w r L 1 T (L p 1 ) + t1 1/r 2 2 w ) r L 2 T (L p ) + C 0t.

23 The uniqueness: Lagrangian formulation We define the flow ψ associated to the velocity field u to be the solution of ψ(t, y) := y + where t 0 u ( τ, ψ(τ, y) ) dτ = y + f (t, y) := f ( t, ψ(t, y) ). We set J := det ( Dψ ) and A := ( Dψ ) 1. We use D x f = D y f A x K = J 1 ( ) div y adj Dψ K div x H = J 1 ( ) div y adj Dψ H. t f + div (f u) = J 1 ( ) t J f. t 0 u(τ, y) dτ.

24 2) Uniqueness Lagrangian coordinates (!) We do not use the formulation in (ρ, w) = equations for (ρ, u) (ρ, u) : { J ρ = ρ0 ρ 0 t u + L u = div ( adj Dψ P(J 1 ρ 0 ) ) = equation for δu = u 1 u 2 : ρ 0 t δu + Lδu = (L L ) 1 δu + δl u 2 div ( δadj P(J 1 1 ρ 0 ) ) ( ( div adj Dψ 2 P(J 1 1 ρ 0 ) P(J 1 2 ρ 0 ) )), where we have set δadj := adj Dψ 1 adj Dψ 2.

25 We need to control δadj, δa and δj: adj Dψ 1 (t) adj Dψ 2 (t) L p J ±1 1 A 1 (t) A 2 (t) L p C C (t) J±1 2 (t) L p C t 0 t 0 t Stability estimates in the energy space d dt ρ 0 δu 2 dx+ δu 2 dx C t 0 δu(τ) L p δu(τ) L p dτ dτ δu(τ) L p dτ. (1 + u 2 ) 2L t δu 2 L dτ 2 0 (!) We need to control t w L t u L L 2 ([0,T]) L2 ([0,T]) = maximal regularity with time weights ( Huang & P. & Zhang 2013 )

26 Propagation of singularities Corollary ( Danchin & Fanelli & P ) Ω 0 R n bounded of class W 2,p ρ 0 = c 1 1 Ω0 + c 2 1 Ω c 0, c 1 c 2 ε Hypotheses for existence =! (ρ, u) solution on [0, T] R n. Moreover: (i) Ω t := ψ t (Ω 0 ) still of class W 2,p (ii) ρ(t) has a jump discontinuity across Ω t

27 Let Ω 0 be a simply connected bounded domain in R 2 such that Ω 0 = f 1 0 ({0}), f 0 W 2,p (R 2 ), f 0 0 on Ω 0 and X 0 = f 0. Let Ω(t) = ψ(t, Ω 0 ), with the ψ the flow of u. The boundary Ω(t) is the level surface of f (t, ), f (t, ψ(t, )) = f 0 ( ) Let X 0 (t, ) = f (t, ), X 1 (t, ) = (1 α) 1, X 2 (t, ) = (1 α) 2. Then X ρ 0 L p, u 0 B 2 2 r p,r, using the theorem we get u L 1 (L ) and X(t, ) W 1,p and finally Ω t W 2,p.

28 Striated Regularity of 2-D incompressible inhomogeneous Navier-Stokes with variable viscosity joint work with Ping Zhang

29 We investigate the global existence and uniqueness of strong solutions to the two-dimensional incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. We study the propagation of regularity for the interface between fluids with different densities with viscous coefficient depending on the density. t ρ + div (ρu) = 0, (t, x) R + R 2, t (ρu) + div (ρu u) div ( µ(ρ)( u + T u) ) + Π = 0, div u = 0, where the viscous coefficient µ(ρ) is a smooth non-decreasing positive function on [0, ). Such a system describes a fluid which is obtained by mixing several immiscible fluids that are incompressible and that have different densities. It may also describe a fluid containing a melted substance.

30 Theorem ( P. & Zhang 2007 ) Let p > 2, X = ( X λ ) Ẇ 1,p (R 2 ) L be a non-degenerate family of vector fields. Let u 0 Ḣ 2δ (R 2 ) H 1 (R 2 ), δ ]1/p, 1/2[, and ρ 0 L (R 2 ) with ρ 0 1 L 2 (R 2 ) and Then, if Xλ µ(ρ 0 ) L C µ,x µ(ρ 0 ) 1 L ɛ 0, m ρ 0 (x) M (INS) has unique global solution u L (R + ; H 1 (R 2 )) L 1 (R + ; Lip(R 2 )). Moreover, X λ (t) L (R + ; W 1,p (R 2 )) and Xλ (t)µ(ρ) L t (L ) C µ,x.

31 We use energy estimates in order to obtain D t u L 2 t (H 1 (R 2 )) New difficulties to obtain u L 1 (L ) We can not solve div (µ(ρ)du) + Π = ρd t u L 2 t (H 1 ) for only bounded coefficients If and X µ(ρ) then solve the elliptic problem: u + Π = div ((µ(ρ) 1)Du) ρd t u u = 1 Pdiv ((µ(ρ) 1)Du) + 1 P(ρD t u) Using again the tangential regularity, we obtain u L C ( ) I(X) µ(ρ) 1 L u L + X u L p). Energy estimates to control X u L p.

32 THANK YOU!