Viscous capillary fluids in fast rotation · Capillary ﬂuids: the model Navier-Stokes-Korteweg...

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


Contents of the talk

Introduction: the model

Navier-Stokes-Korteweg with Coriolis force

(i) Results

(ii) Sketch of the proof

(iii) Final remarks



COMPRESSIBLE FLUIDS

WITH

CAPILLARITY EFFECTS



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



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 )



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



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



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



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 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



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)



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



Constant capillarity regime: α = 0


)ε



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



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



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



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


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



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



=⇒ 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



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



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


)ε



a) div u ≡ 0

b) u =(uh(xh) , 0

), with c(xh) uh = ∇⊥h

(Id −∆h

)r

c) r solves a linear “parabolic” equation



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



THANK YOU !


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Viscous capillary fluids in fast rotation · Capillary ﬂuids: the model Navier-Stokes-Korteweg...

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