Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Viscous capillary fluids in fast rotation
Francesco Fanelli
Centro di Ricerca Matematica “Ennio De Giorgi”
SCUOLA NORMALE SUPERIORE
BCAM – BASQUE CENTER FOR APPLIED MATHEMATICS
BCAM Scientific Seminar
Bilbao – May 19, 2015
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Contents of the talk
Introduction: the model
Navier-Stokes-Korteweg with Coriolis force
(i) Results
(ii) Sketch of the proof
(iii) Final remarks
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
COMPRESSIBLE FLUIDS
WITH
CAPILLARITY EFFECTS
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
The general system
∂tρ + 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 density of the fluid
u(t, x) ∈ R3 velocity field
Π(ρ) = ργ / γ pressure of the fluid (γ ≥ 1)
Du := (1/2)(∇u + t∇u
)κ > 0 capillarity coefficient
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
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 )
B Well-posedness in critical Besov spaces
( Danchin & Desjardins – 2001 )
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Navier-Stokes-Korteweg system
∂tρ+ div (ρu) = 0
∂t (ρu) + div(ρu⊗ u
)+∇Π(ρ)− ν0 div
(ρDu
)− κ ρ∇∆ρ = 0
Capillarity term:
κ > 0 and σ(ρ) = ρ
Viscosity cofficients:
ν(ρ) = ν0ρ and λ(ρ) ≡ 0
B Degeneracy for ρ ∼ 0
B Surface tension control on ∇2ρ
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Theorem ( Bresch & Desjardins & Lin – 2003 )
∃ global in time “weak” solutions (ρ, u)
Remarks
(i) Domain: Ω = Td ( d = 2 , 3 ) or
= Td−1× ]0, 1[
(ii) Weak solutions à la Leray
(iii) “Weak”: momentum equation tested on ρϕ , ϕ ∈ D(Ω)∫ T
0
∫Ω
(ρ2u · ∂tϕ+ ρ2u⊗ u : ∇ϕ− ρ2u · ϕdivu−
−νρ2D(u) : ∇ϕ− νρD(u) : ϕ⊗∇ρ+ Π(ρ)ρdivϕ−
−κρ2∆ρdivϕ− 2κρ∇ρ · ϕ∆ρ)
dxdt = −∫
Ωρ2
0u0 · ϕ(0)dx
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
On the proof
1) A priori estimates
B Classical energy
=⇒ ρ ∈ L∞T Lγ , ∇ρ ∈ L∞
T L2 ,√ρ u ∈ L∞
T L2 ,√ρDu ∈ L2
TL2
B BD entropy
=⇒ ∇2ρ ∈ L2TL2 , ∇√ρ ∈ L∞
T L2
2) Construction of smooth approximated solutions(ρn , un
)n
3) Stability analysis
B Compactness of(ρ
3/2n un
)n in L2
TL2loc
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
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 )
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
NAVIER-STOKES-KORTEWEG
WITH
CORIOLIS FORCE
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Fluid models with Coriolis force
Motivation: description of large scale phenomenaB quantitative aspectsB qualitative aspects ( physical effects )
General hypotheses:(i) Rotation around the vertical axis x3
(ii) Constant rotation speed=⇒ rotation operator: u 7→
(e3 × u
)/Ro
(iii) Complete slip boundary conditions=⇒ NO boundary layers effects
Singular perturbation problem: Ro ∼ ε
=⇒ asymptotic behavior of weak solutions for ε → 0
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
N-S-K with Coriolis force
∂tρ + div(ρ u)
= 0
∂t(ρ u)
+ div(ρ u⊗ u
)+
1ε2 ∇Π(ρ) +
+e3 × ρ u
ε− ν div
(ρDu
)− 1ε2(1−α)
ρ∇∆ρ = 0
Ω = R2× ]0, 1[ + complete slip boundary conditions
Π(ρ) = ρ2 / 2
Mach number ∼ ε and Rossby number ∼ ε
κ ∼ ε2α , with 0 ≤ α ≤ 1
B Ill-prepared initial data(i) ρ0,ε = 1 + ε r0,ε , with
(r0,ε)ε⊂ H1(Ω) ∩ L∞(Ω)
(ii)(u0,ε)ε⊂ L2(Ω)
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Statements
Vanishing capillarity limit: 0 < α ≤ 1
Theorem ( F. – 2014 )(ρε , uε
)ε
weak solutions, ρε = 1 + ε rε
rε r ,√ρε uε u
a) div u ≡ 0
b) u =(uh(xh) , 0
), with uh = ∇⊥h r
c) r solves a quasi-geostrophic equation
∂t (r − ∆hr) + ∇⊥h r · ∇h∆hr + ν∆2hr = 0
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Constant capillarity regime: α = 0
Theorem ( F. – 2014 )(ρε , uε
)ε
weak solutions, ρε = 1 + ε rε
rε r ,√ρε uε u
a) div u ≡ 0
b) u =(uh(xh) , 0
), with uh = ∇⊥h
(Id −∆h
)r
c) r solves
∂t
((Id −∆h + ∆2
h)r)
+
+∇⊥h(Id −∆h
)r · ∇h∆2
hr + ν∆2h(Id −∆h
)r = 0
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Related results
2-D viscous shallow water with friction terms( Bresch & Desjardins – 2003 ) Viscosity = − ν div
(ρDu
), capillarity = − ρ∇∆ρ
General Navier-Stokes-Korteweg system( Jüngel & Lin & Wu – 2014 ) Viscous tensor − div
(ν(ρ) Du
) Capillarity term −κ ρ∇
(σ′(ρ) ∆σ(ρ)
) Strong solutions framework; local in time study
B Incompressible + high rotation + vanishing capillarity
B Ω = T2
B Well-prepared initial data, modulated energy method
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Remarks
Weak solutions in the sense of Bresch–Desjardins–Lin:
momentum equation tested on ρε ϕ , ϕ ∈ D(Ω)
Constant capillarity: more general pressure laws
Π(ρ) = ργ / γ , with 1 < γ ≤ 2
B Problem for 0 < α ≤ 1 : BD entropy estimates
Vanishing capillarity
B Uniqueness criterion for the limit equation
B 0 < α < 1 =⇒ anisotropy of scaling
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Main steps of the proof
(i) Uniform bounds
a. Classical energy conservation
b. BD entropy
B Control of the rotation term uniformly in ε
→ Control local in time
→ Necessary to have ‖ρε − 1‖L∞T L2 ∼ O(ε)
(ii) Constraint on the limit
a.√ρε uε u ,
√ρε Duε U , with U = Du
b. Taylor-Proudman theorem + stream-function relation
(iii) Propagation of acoustic waves
B Spectral analysis ( Feireisl & Gallagher & Novotný – 2012 )Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Ruelle-Amrein-Georgescu-Enss theorem
RAGE theorem
B : D(B) ⊂ H −→ H self-adjoint on H Hilbert
H = Hcont ⊕ Eigen (B)
Πcont := orthogonal projection onto Hcont
K : H −→ H compact
=⇒ for T −→ +∞ ,∥∥∥∥ 1T
∫ T
0e−i tB K Πcont ei tB dt
∥∥∥∥L(H)
−→ 0
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
End of the proof for α = 1
Acoustic propagator A :
(r
V
)7−→
(div V
e3 × V + ∇r
)
=⇒ system ε ∂t
(rε
Vε
)+ A
(rε
Vε
)= ε
(0
Fε
)
K : L2(Ω) × L2(Ω) −→ KerA orthogonal projection
(i) K[rε,Vε] strongly converges in L2TL2
loc
(ii) σp(A) = 0 RAGE theorem
=⇒ K⊥[rε,Vε] → 0 strongly in L2TL2
loc
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
=⇒ Strong convergence of(rε)ε
and(ρ
3/2ε uε
)ε
in L2TL2
loc
=⇒ Passing to the limit
B Constant capillarity: α = 0
A A0
(r
V
):=
(div V
e3 × V + ∇(Id − ∆
)r
)
B Symmetrization of the system + RAGE theorem
Microlocal symmetrizer:
〈(r1,V1) , (r2,V2)〉0 := 〈r1 , (Id −∆)r2〉L2 + 〈V1 , V2〉L2
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Anisotropic scaling: 0 < α < 1
Singular perturbation operator:
A(α)ε
(r
V
):=
(div V
e3 × V + ∇(Id − ε2α∆
)r
)
System: ε ∂t
(rε
Vε
)+ A(α)
ε
(rε
Vε
)= ε
(0
Fε,α
)
=⇒ adapted version of the RAGE theorem
B Changing operators and metrics
B σp(A(α)ε ) = 0
B Operators and metrics are linked
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Variable rotation axis
Coriolis operator C(ρ, u) = c(xh) e3 × ρ u
(i) c has non-degenerate critical points
(ii) ∇hc ∈ Cµ(R2) for µ = admissible modulus of continuity
Theorem ( F. – 2015 )(ρε , uε
)ε
weak solutions, ρε = 1 + ε rε
rε r ,√ρε uε u
a) div u ≡ 0
b) u =(uh(xh) , 0
), with c(xh) uh = ∇⊥h
(Id −∆h
)r
c) r solves a linear “parabolic” equation
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
Remarks
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(xh)
Zygmund conditions
Francesco Fanelli Navier-Stokes-Korteweg with rotation
Capillary fluids: the modelNavier-Stokes-Korteweg with Coriolis
THANK YOU !
Francesco Fanelli Navier-Stokes-Korteweg with rotation