Viscous capillary fluids in fast rotation

Transcription

1 Viscous capillary fluids in fast rotation Centro di Ricerca Matematica Ennio De Giorgi SCUOLA NORMALE SUPERIORE BCAM BASQUE CENTER FOR APPLIED MATHEMATICS BCAM Scientific Seminar Bilbao May 19, 2015

2 Contents of the talk Capillary fluids: the model Introduction: the model force (i) Results (ii) Sketch of the proof (iii) Final remarks

3 COMPRESSIBLE FLUIDS WITH CAPILLARITY EFFECTS

4 The general system Capillary fluids: the model t ρ + div (ρu) = 0 t (ρu) + div ( ρu u ) + Π(ρ) = = div ( ν(ρ)du + λ(ρ)div u Id ) + κρ ( σ (ρ) σ(ρ) ) ρ(t, x) 0 u(t, x) R 3 density of the fluid velocity field Π(ρ) = ρ γ / γ pressure of the fluid (γ 1) Du := (1/2) ( u + t u ) κ > 0 capillarity coefficient

5 κ = 0, ν(ρ) = ν > 0, λ(ρ) = λ, ν + λ > 0 = existence of global weak solutions ( P.-L. Lions 1993 ) κ > 0, σ(ρ) = ρ, ν > 0, λ = ν/3 = local existence of strong solutions global if initial data close to a stable equilibrium ( Hattori & Li 1996 ) Well-posedness in critical Besov spaces ( Danchin & Desjardins 2001 )

6 Navier-Stokes-Korteweg system t ρ + div (ρu) = 0 t (ρu) + div ( ρu u ) + Π(ρ) ν 0 div ( ρ Du ) κ ρ ρ = 0 Capillarity term: κ > 0 and σ(ρ) = ρ Viscosity cofficients: ν(ρ) = ν 0 ρ and λ(ρ) 0 Degeneracy for ρ 0 Surface tension control on 2 ρ

7 Theorem ( Bresch & Desjardins & Lin 2003 ) global in time weak solutions (ρ, u) Remarks (i) Domain: Ω = T d ( d = 2, 3 ) or (ii) Weak solutions = T d 1 ]0, 1[ à la Leray (iii) Weak : momentum equation tested on ρ ϕ, ϕ D(Ω) T ( ρ 2 u t ϕ + ρ 2 u u : ϕ ρ 2 u ϕdivu 0 Ω νρ 2 D(u) : ϕ νρd(u) : ϕ ρ + Π(ρ)ρdivϕ ) κρ 2 ρdivϕ 2κρ ρ ϕ ρ dxdt = ρ 2 0u 0 ϕ(0)dx Ω

8 On the proof Capillary fluids: the model 1) A priori estimates Classical energy = ρ L T L γ, ρ L T L 2, ρ u L T L 2, ρ Du L 2 TL 2 BD entropy = 2 ρ L 2 TL 2, ρ L T L 2 2) Construction of smooth approximated solutions ( ρ n, u n )n 3) Stability analysis Compactness of ( ρ 3/2 n u n )n in L 2 TL 2 loc

9 On the BD entropy structure 2-D viscous shallow water + friction terms ( Bresch & Desjardins 2003 ) Compressible Navier-Stokes with heat conduction ( Bresch & Desjardins 2007 ) 1-D lubrication models with strong slippage ( Kitavtsev & Laurençot & Niethammer 2011 ) Barotropic compressible Navier-Stokes ( Mellet & Vasseur 2007 ) Singular pressure laws ( Bresch & Desjardins & Zatorska 2015 )

10 NAVIER-STOKES-KORTEWEG WITH CORIOLIS FORCE

11 Fluid models with Coriolis force Motivation: description of large scale phenomena quantitative aspects qualitative aspects ( physical effects ) General hypotheses: (i) Rotation around the vertical axis x 3 (ii) Constant rotation speed = rotation operator: u ( e 3 u ) / Ro (iii) Complete slip boundary conditions = NO boundary layers effects Singular perturbation problem: Ro ε = asymptotic behavior of weak solutions for ε 0

12 N-S-K with Coriolis force t ρ + div ( ρ u ) = 0 ( ) ( ) 1 t ρ u + div ρ u u + ε 2 Π(ρ) + + e3 ρ u ν div ( ρ Du ) ε 1 ρ ρ = 0 ε2(1 α) Ω = R 2 ]0, 1[ + complete slip boundary conditions Π(ρ) = ρ 2 / 2 Mach number ε and Rossby number ε κ ε 2α, with 0 α 1 Ill-prepared initial data (i) ρ 0,ε = 1 + ε r 0,ε, (ii) ( u 0,ε )ε L2 (Ω) with ( r0,ε ) ε H1 (Ω) L (Ω)

13 Statements Capillary fluids: the model Vanishing capillarity limit: 0 < α 1 Theorem ( F ) ( ) ρε, u ε weak solutions, ρ ε ε = 1 + ε r ε r ε r, ρε u ε u a) div u 0 b) u = ( u h (x h ), 0 ), with u h = h r c) r solves a quasi-geostrophic equation t (r h r) + h r h h r + ν 2 hr = 0

14 Constant capillarity regime: α = 0 Theorem ( F ) ( ) ρε, u ε weak solutions, ρ ε ε = 1 + ε r ε r ε r, ρε u ε u a) div u 0 b) u = ( u h (x h ), 0 ), with u h = ( ) h Id h r c) r solves ) t ((Id h + 2 h)r + + h ( ) Id h r h 2 hr + ν 2 ( ) h Id h r = 0

15 Related results Capillary fluids: the model 2-D viscous shallow water with friction terms ( Bresch & Desjardins 2003 ) Viscosity = ν div ( ρ Du ), General Navier-Stokes-Korteweg system ( Jüngel & Lin & Wu 2014 ) Viscous tensor div ( ν(ρ) Du ) capillarity = ρ ρ Capillarity term κ ρ ( σ (ρ) σ(ρ) ) Strong solutions framework; local in time study Incompressible + high rotation + vanishing capillarity Ω = T 2 Well-prepared initial data, modulated energy method

16 Remarks Capillary fluids: the model Weak solutions in the sense of Bresch Desjardins Lin: momentum equation tested on ρ ε ϕ, ϕ D(Ω) Constant capillarity: more general pressure laws Π(ρ) = ρ γ / γ, with 1 < γ 2 Problem for 0 < α 1 : BD entropy estimates Vanishing capillarity Uniqueness criterion for the limit equation 0 < α < 1 = anisotropy of scaling

17 Main steps of the proof (i) Uniform bounds a. Classical energy conservation b. BD entropy Control of the rotation term uniformly in ε Control local in time Necessary to have ρ ε 1 L T L 2 O(ε) (ii) Constraint on the limit a. ρε u ε u, ρε Du ε U, with U = Du b. Taylor-Proudman theorem + stream-function relation (iii) Propagation of acoustic waves Spectral analysis ( Feireisl & Gallagher & Novotný 2012 )

18 Ruelle-Amrein-Georgescu-Enss theorem RAGE theorem B : D(B) H H self-adjoint on H Hilbert H = H cont Eigen (B) Π cont := orthogonal projection onto H cont K : H H compact = for T +, 1 T T 0 e i t B K Π cont e i t B dt 0 L(H)

19 End of the proof for α = 1 Acoustic propagator A : ( r V ) ( div V ) e 3 V + r = system ε t ( rε V ε ) + A ( rε V ε ) = ε ( ) 0 F ε K : L 2 (Ω) L 2 (Ω) Ker A orthogonal projection (i) K[r ε, V ε ] strongly converges in L 2 TL 2 loc (ii) σ p (A) = {0} RAGE theorem = K [r ε, V ε ] 0 strongly in L 2 TL 2 loc

20 = Strong convergence of ( ) r ε ε and ( ρ 3/2 ε u ε )ε in L2 T L2 loc = Passing to the limit Constant capillarity: α = 0 A A 0 ( r V ) := ( div V ) e 3 V + ( Id ) r Symmetrization of the system + RAGE theorem Microlocal symmetrizer: (r 1, V 1), (r 2, V 2) 0 := r 1, (Id )r 2 L 2 + V 1, V 2 L 2

21 Anisotropic scaling: 0 < α < 1 Singular perturbation operator: ( ) ( ) r A (α) div V ε := V e 3 V + ( Id ε 2α ) r System: ε t ( rε V ε ) + A (α) ε ( rε V ε ) = ε ( ) 0 F ε,α = adapted version of the RAGE theorem Changing operators and metrics σ p (A (α) ε ) = {0} Operators and metrics are linked

22 Variable rotation axis Coriolis operator C(ρ, u) = c(x h ) e 3 ρ u (i) c has non-degenerate critical points (ii) h c C µ (R 2 ) for µ = admissible modulus of continuity Theorem ( F ) ( ) ρε, u ε weak solutions, ρ ε ε = 1 + ε r ε r ε r, ρε u ε u a) div u 0 b) u = ( u h (x h ), 0 ), with c(x h ) u h = ( ) h Id h r c) r solves a linear parabolic equation

23 Remarks Capillary fluids: the model 1) Singular perturbation operator: variable coefficients = compensated compactness arguments Gallagher & L. Saint-Raymond 2006 Feireisl & Gallagher & Gérard-Varet & Novotný ) Novelties: Surface tension term Less regularity available for the approximation 3) Regularity of c(x h ) Zygmund conditions

24 THANK YOU!