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
Contents of the talk Capillary fluids: the model Introduction: the model force (i) Results (ii) Sketch of the proof (iii) Final remarks
COMPRESSIBLE FLUIDS WITH CAPILLARITY EFFECTS
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
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 )
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
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
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
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 )
NAVIER-STOKES-KORTEWEG WITH CORIOLIS FORCE
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
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)
Statements Capillary fluids: the model Vanishing capillarity limit: 0 < α 1 Theorem ( F. 2014 ) ( ) 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
Constant capillarity regime: α = 0 Theorem ( F. 2014 ) ( ) 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
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
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
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 )
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)
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
= 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
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
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. 2015 ) ( ) 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
Remarks Capillary fluids: the model 1) Singular perturbation operator: variable coefficients = compensated compactness arguments Gallagher & L. Saint-Raymond 2006 Feireisl & Gallagher & Gérard-Varet & Novotný 2012 2) Novelties: Surface tension term Less regularity available for the approximation 3) Regularity of c(x h ) Zygmund conditions
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