Boundary layers for Navier-Stokes equations with slip ......Uniform existence and convergence towards Euler Improvements and possible extensions 2 / 40 Matthew Paddick Boundary layers

Stability and instability in the 2D incompressible modelConvergence of strong solutions in the 3D isentropic model

Boundary layers for Navier-Stokes equations withslip boundary conditions

Matthew Paddick

Laboratoire Jacques-Louis LionsUniversite Pierre et Marie Curie (Paris 6)

Seminaire EDP, Universite Paris-Est Creteil16th April 2015

1 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions


1 Stability and instability in the 2D incompressible modelThe inviscid limit problem and boundary layersNavier boundary conditions and L2 stabilityL∞ instability of shear flows

2 Convergence of strong solutions in the 3D isentropic modelThe equations and conormal derivativesUniform existence and convergence towards EulerImprovements and possible extensions



The inviscid limit problem and boundary layersNavier boundary conditions and L2 stabilityL∞ instability of shear flows






2D incompressible Navier-Stokes on the half-space(x , y) ∈ Ω = R×]0,+∞[: ∂tuε + uε · ∇uε − ε∆uε +∇pε = 0

div uε = 0uε|t=0 = uε0 ,

(1)

modelling the motion of a viscous, homogeneous fluid.

Usual boundary conditions (BC) for macroscopic flows:

impermeability of the boundary

u2|y=0 = 0,

and the fluid does not slip along the boundary - Dirichlet BC

u|y=0 = 0.




Nondimensional viscosity ε = 1/Re, Re the Reynolds number.

Inviscid limit problem: behaviour of the solutions uε relative to v ,solution of the incompressible Euler equation ∂tv + v · ∇v +∇q = 0

div v = 0v |t=0 = v0

(2)

when ε goes to 0 ⇔ high-Reynolds approximation.

The Euler equation is of order one, so only one BC is required:

v2|y=0 = 0.




As Navier-Stokes solutions can converge to Euler solutions in the wholespace, a Prandtl boundary layer expansion1 is proposed:

uε(t, x , y) ∼ε→0

v(t, x , y) + V

(t, x ,

y√ε

)(3)

Smithsonian National Air and Space Museum

Boundary layer stability in a normed function space (E , ‖·‖) if

supt∈[0,T ]

‖uε(t)− v(t)‖ ε→0−→ 0.

1Prandtl, Verhand. III Intern. Math. Kongresses Heidelberg, 19046 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



As Navier-Stokes solutions can converge to Euler solutions in the wholespace, a Prandtl boundary layer expansion1 is proposed:

uε(t, x , y) ∼ε→0

v(t, x , y) + V

(t, x ,

y√ε

)(3)

Smithsonian National Air and Space Museum

Boundary layer stability in a normed function space (E , ‖·‖) if

supt∈[0,T ]

‖uε(t)− v(t)‖ ε→0−→ 0.

1Prandtl, Verhand. III Intern. Math. Kongresses Heidelberg, 19046 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Issues with the inviscid limit problem with Dirichlet BCs:

if u = (u1, u2)(t, x , y) = (u,√εv)(t, x , y ′ = ε−1/2y), the boundary

layer is governed by the ill-posed2 Prandtl equation:∂t u + u∂x u + v∂y ′ u − ∂y ′y ′ u = (∂tv + v · ∇v)1|y=0

∂x u + ∂y ′ v = 0(u, v)|y=0 = 0

limy→+∞ u = v1|y=0;

even if Prandtl is well-posed, ansatz (3) can be invalid due tocreation of vorticity inside the boundary layer3, and control of thevorticity (turbulence) near the boundary is crucial4.

2Gerard-Varet & Dormy, J. Amer. Math. Soc., 20103Grenier, Comm. Pure Appl. Math., 20004Kato, Seminar on nonlinear PDEs, 1984




Issues with the inviscid limit problem with Dirichlet BCs:

if u = (u1, u2)(t, x , y) = (u,√εv)(t, x , y ′ = ε−1/2y), the boundary

layer is governed by the ill-posed2 Prandtl equation:∂t u + u∂x u + v∂y ′ u − ∂y ′y ′ u = (∂tv + v · ∇v)1|y=0

∂x u + ∂y ′ v = 0(u, v)|y=0 = 0

limy→+∞ u = v1|y=0;

even if Prandtl is well-posed, ansatz (3) can be invalid due tocreation of vorticity inside the boundary layer3, and control of thevorticity (turbulence) near the boundary is crucial4.

2Gerard-Varet & Dormy, J. Amer. Math. Soc., 20103Grenier, Comm. Pure Appl. Math., 20004Kato, Seminar on nonlinear PDEs, 1984




Our boundary condition: the Navier, or slip, boundary condition5, amixed BC that allows the fluid to slip along the boundary at a speedproportional to the normal stress:

u2|y=0 = 0 and ∂y uε1 |y=0 = 2aεuε1 |y=0 (4)

The slip phenomenon is observed at microscopic levels, for instance incapillary blood vessels.

5Navier, Mem. Acad. Roy. Sci. Inst. France, 18238 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Let aε = a/εβ for a, β ∈ R. β > 0: approximation of Dirichlet BCs asε→ 0, transition between the stable β = 0 case and the unstablevorticity in the Dirichlet (aε =∞) case.With a > 0 and β = 0: stability in L2, using energy estimates6.

Weextend this result:

Theorem 1.1 (Diff. Integral Eqns. 2014)

Let uε0 be a converging sequence in L2(Ω), whose limit v0 is in Hs(Ω) forsome s > 2. Then, for every time T > 0, the sequence of global weaksolutions to the Navier-Stokes equation uε with initial condition uε0 andNavier boundary condition as above, converges in L∞([0,T ], L2(Ω))towards the unique strong solution v of the Euler equation with initialcondition v0 if

either a > 0 and β < 1,

or a < 0 and β ≤ 1/2.

6Iftimie & Planas, Nonlinearity, 20069 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Let aε = a/εβ for a, β ∈ R. β > 0: approximation of Dirichlet BCs asε→ 0, transition between the stable β = 0 case and the unstablevorticity in the Dirichlet (aε =∞) case.With a > 0 and β = 0: stability in L2, using energy estimates6. Weextend this result:


Let uε0 be a converging sequence in L2(Ω), whose limit v0 is in Hs(Ω) forsome s > 2. Then, for every time T > 0, the sequence of global weaksolutions to the Navier-Stokes equation uε with initial condition uε0 andNavier boundary condition as above, converges in L∞([0,T ], L2(Ω))towards the unique strong solution v of the Euler equation with initialcondition v0 if

either a > 0 and β < 1,

or a < 0 and β ≤ 1/2.

6Iftimie & Planas, Nonlinearity, 20069 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Further extensions.

The theorem can be extended to higher dimensions by starting withthe Leray energy inequality.

The first point of the theorem (with a ≥ 0) has been extended tothe isentropic compressible model by Sueur7.

7Sueur, J. Math. Fluid Mech., 201410 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Further extensions.

The theorem can be extended to higher dimensions by starting withthe Leray energy inequality.

The first point of the theorem (with a ≥ 0) has been extended tothe isentropic compressible model by Sueur7.

7Sueur, J. Math. Fluid Mech., 201410 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions








Set a > 0 and β = 1/2: (4) is written

∂y u1|y=0 =2a√ε

u1|y=0. (5)

Shear flow: vector field of the form ush(y) = (ush(y), 0); stationarysolution to the incompressible Euler equation (with constant pressure).It is linearly unstable if there exist λ ∈ C with <(λ) > 0, k ∈ R andψ ∈ H1

0 (R+) such that u(t, x , y) = eλt∇⊥(e ikxψ(y)) solves the Eulerequation linearised around ush: ∂tu + u · ∇ush + ush · ∇u +∇p = 0

div u = 0u2|y=0 = 0,

(6)

For the Euler equation, linear instability implies nonlinear instability8.

8Grenier, Comm. Pure Appl. Math., 200012 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Set a > 0 and β = 1/2: (4) is written

∂y u1|y=0 =2a√ε

u1|y=0. (5)

Shear flow: vector field of the form ush(y) = (ush(y), 0); stationarysolution to the incompressible Euler equation (with constant pressure).It is linearly unstable if there exist λ ∈ C with <(λ) > 0, k ∈ R andψ ∈ H1

0 (R+) such that u(t, x , y) = eλt∇⊥(e ikxψ(y)) solves the Eulerequation linearised around ush: ∂tu + u · ∇ush + ush · ∇u +∇p = 0

div u = 0u2|y=0 = 0,

(6)

For the Euler equation, linear instability implies nonlinear instability8.

8Grenier, Comm. Pure Appl. Math., 200012 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions




Let Y = y/√ε. We consider a family of boundary-layer scale initial

conditions written as ush(Y ), with ush a linearly unstable shear profile forthe Euler equation. We generate a reference solution to theNavier-Stokes equation as follows: ∂tush(t,Y )− ∂2

YY ush(t,Y ) = 0ush(0,Y ) = ush(Y )

∂Y ush(t, 0) = 2aush(t, 0)

Then, for any n ∈ N∗, there exist δ0 > 0 and ε0 > 0 such that, for every0 < ε < ε0, there exist initial data uε0 such that

‖uε0(x , y)− ush(Y )‖Hs (Ω) ≤ Cεn

for some s > 0, which generates a solution to the Navier-Stokes equationwith boundary condition (5) that satisfies, at a time T ε ∼ −n

√ε ln(ε),

‖uε(T ε, x , y)− ush(T ε,Y )‖L∞(Ω) ≥ δ0.





Let Y = y/√ε. We consider a family of boundary-layer scale initial

conditions written as ush(Y ), with ush a linearly unstable shear profile forthe Euler equation. We generate a reference solution to theNavier-Stokes equation as follows: ∂tush(t,Y )− ∂2

YY ush(t,Y ) = 0ush(0,Y ) = ush(Y )

∂Y ush(t, 0) = 2aush(t, 0)

Then, for any n ∈ N∗, there exist δ0 > 0 and ε0 > 0 such that, for every0 < ε < ε0, there exist initial data uε0 such that

‖uε0(x , y)− ush(Y )‖Hs (Ω) ≤ Cεn

for some s > 0, which generates a solution to the Navier-Stokes equationwith boundary condition (5) that satisfies, at a time T ε ∼ −n

√ε ln(ε),

‖uε(T ε, x , y)− ush(T ε,Y )‖L∞(Ω) ≥ δ0.




Main ideas of the proof. After rescaling the equation(t, x , y) = ε−1/2(t, x , y), construct an approximate solution to theNavier-Stokes equation written as:

uap(t, x , y) = ush(√εt, y) +

N∑j=1

εjnU ij (t, x , y) + εjn+1/4Ub

j (t, x , y/ε1/4).

(7)Note the ε1/4 amplitude of the boundary layer terms, hinted by theasymptotic expansion for fixed Navier BCs9. It will allow to bound ∇uap

independently of ε.Each U i

j and Ubj are built using another asymptotic expansion in powers

of ε1/8, e.g. U i1 =

∑8n−1m=0 ε

m/8uim.

9Iftimie & Sueur, Arch. Ration. Mech. Anal., 201114 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



The first term ui0 is chosen as a wavepacket of unstable solutions of (6):

ui0(t, x , y) =

∫Rϕ0(k)eλ(k)t+ikx v0(k , y) dk (8)

with ϕ0 ∈ C∞0 (R) localising the eigenmodes around the most unstablewavenumber.

Denote σ0 = max<λ(k) | k ∈ R the most unstable eigenvalue, and letgε(t) = εn(1 + t)−1/2eσ0t .

Time of study: t ≤ T ε0 − τ such that gε(T ε

0 ) = 1, and τ > 0independent of ε.




The first term ui0 is chosen as a wavepacket of unstable solutions of (6):

ui0(t, x , y) =

∫Rϕ0(k)eλ(k)t+ikx v0(k , y) dk (8)

with ϕ0 ∈ C∞0 (R) localising the eigenmodes around the most unstablewavenumber.

Denote σ0 = max<λ(k) | k ∈ R the most unstable eigenvalue, and letgε(t) = εn(1 + t)−1/2eσ0t .

Time of study: t ≤ T ε0 − τ such that gε(T ε

0 ) = 1, and τ > 0independent of ε.




The first term of uap carries the growth we are after:∥∥ui0(t)

∥∥L2(ΩA(t))

≥ C ′1(1 + t)1/4gε(t),

for ΩA(t) ⊂ Ω of measure√

1 + t.

By induction, we then show that∥∥∥εjn(U ij (t) + ε1/4Ub

j (t))∥∥∥

L2(Ω)≤ Cj (1 + t)1/4(gε(t))j .

This is done by noticing that the interior terms U ij solve (6) with a source

term Fj satisfying

‖Fj (t)‖Hs (Ω) ≤ C ′jeγt

(1 + t)α

with γ > σ0 and α > 0. A resolvent estimate for the linearised equationthen yields that U i

j also satisfies this inequality.




The first term of uap carries the growth we are after:∥∥ui0(t)

∥∥L2(ΩA(t))

≥ C ′1(1 + t)1/4gε(t),

for ΩA(t) ⊂ Ω of measure√

1 + t.

By induction, we then show that∥∥∥εjn(U ij (t) + ε1/4Ub

j (t))∥∥∥

L2(Ω)≤ Cj (1 + t)1/4(gε(t))j .

This is done by noticing that the interior terms U ij solve (6) with a source

term Fj satisfying

‖Fj (t)‖Hs (Ω) ≤ C ′jeγt

(1 + t)α

with γ > σ0 and α > 0. A resolvent estimate for the linearised equationthen yields that U i

j also satisfies this inequality.




Now we examine w = uε − uap. Choosing N so that

Nσ0 > ‖∇uap‖L∞([0,Tε0 −τ ]×Ω) + 1/2,

a quick energy estimate on the equation satisfied by w yields

‖w(t)‖L2(Ω) ≤ CN+1(1 + t)1/4(gε(t))N+1.

As a result,

∥∥uε(t)− ush(√εt)∥∥

L2(ΩA(t))≥ (1 + t)1/4

C ′1gε(t)−N+1∑j=2

Cj (gε(t))j

≥ (1 + t)1/4(C ′1/2)gε(t)

for t ≤ T ε0 − τ ′, with τ ′ > τ independent of ε.





Nσ0 > ‖∇uap‖L∞([0,Tε0 −τ ]×Ω) + 1/2,


‖w(t)‖L2(Ω) ≤ CN+1(1 + t)1/4(gε(t))N+1.

As a result,


L2(ΩA(t))≥ (1 + t)1/4

C ′1gε(t)−N+1∑j=2

Cj (gε(t))j

≥ (1 + t)1/4(C ′1/2)gε(t)






Nσ0 > ‖∇uap‖L∞([0,Tε0 −τ ]×Ω) + 1/2,


‖w(t)‖L2(Ω) ≤ CN+1(1 + t)1/4(gε(t))N+1.

As a result,


L2(ΩA(t))≥ (1 + t)1/4

C ′1gε(t)−N+1∑j=2

Cj (gε(t))j

≥ (1 + t)1/4(C ′1/2)gε(t)






Nσ0 > ‖∇uap‖L∞([0,Tε0 −τ ]×Ω) + 1/2,


‖w(t)‖L2(Ω) ≤ CN+1(1 + t)1/4(gε(t))N+1.

As a result,


L2(ΩA(t))≥ (1 + t)1/4

C ′1gε(t)−N+1∑j=2

Cj (gε(t))j

≥ (1 + t)1/4δ0

for t = T ε0 − T , with T > τ ′ chosen independently of ε.




Comparison with the Dirichlet boundary condition.

WKB expansion based on the Prandtl ansatz (3): O(1) boundarylayers.

The study time T ε0 is shorter. One misses instability in L∞, ε1/4δ0

replaces δ0 in the final equality, but we do get instability of thevorticity.




Robustness of the method.This ‘linear instability implies nonlinear instability’ argument requires:

a ‘most unstable’ eigenmode for the linearised system,

a resolvent estimate for the linearised system,

and good energy estimates for the equation solved by the differencebetween the exact and approximate solutions.

Thus it has also been used to obtain transverse instability of solitarywaves in numerous settings: NLS, KP-I, water-waves10,Euler-Korteweg11,...

10Rousset & Tzvetkov, multiple references11arxiv 1502.05647




Robustness of the method.This ‘linear instability implies nonlinear instability’ argument requires:

a ‘most unstable’ eigenmode for the linearised system,

a resolvent estimate for the linearised system,

and good energy estimates for the equation solved by the differencebetween the exact and approximate solutions.

Thus it has also been used to obtain transverse instability of solitarywaves in numerous settings: NLS, KP-I, water-waves10,Euler-Korteweg11,...

10Rousset & Tzvetkov, multiple references11arxiv 1502.05647



The equations and conormal derivativesUniform existence and convergence towards EulerImprovements and possible extensions






3D isentropic, compressible Navier-Stokes equation on the half spaceΩ = X = (x , y , z) ∈ R2×]0,+∞[:

∂tρ+ div (ρu) = 0ρ∂tu + ρu · ∇u = div Σ + ρF ,

(9)

modelling the motion of a viscous, non-heat-conducting fluid withtypically supersonic speeds.

Σ is the stress tensor:

Σ = εµ(ρ)(∇+∇t)u + ελ0(ρ)div uI3 − P(ρ)I3. (10)

The viscosity coefficients µ > 0 and µ+ λ0 > 0 can dependsmoothly on the density.

The pressure P(ρ) is given by a barotropic law: P(ρ) = kργ withk > 0 and γ > 1.




3D isentropic, compressible Navier-Stokes equation on the half spaceΩ = X = (x , y , z) ∈ R2×]0,+∞[:

∂tρ+ div (ρu) = 0ρ∂tu + ρu · ∇u = div Σ + ρF ,

(9)

modelling the motion of a viscous, non-heat-conducting fluid withtypically supersonic speeds.

Σ is the stress tensor:

Σ = εµ(ρ)(∇+∇t)u + ελ0(ρ)div uI3 − P(ρ)I3. (10)

The viscosity coefficients µ > 0 and µ+ λ0 > 0 can dependsmoothly on the density.

The pressure P(ρ) is given by a barotropic law: P(ρ) = kργ withk > 0 and γ > 1.




F is a smooth force term.We solve for t ∈ R with the fluid at rest for negative times:

(ρ, u,F ) = (1, 0, 0) for t < 0.

Navier boundary condition:

u3|z=0 = 0 and (ε−1Σ~n + au)τ |z=0 = 0,

where vτ = (v1, v2) is the tangential part of a 3D vector v , anda ∈ R.




F is a smooth force term.We solve for t ∈ R with the fluid at rest for negative times:

(ρ, u,F ) = (1, 0, 0) for t < 0.

Navier boundary condition:

u3|z=0 = 0 and [µ(ρ)∂z uτ ]|z=0 = 2auτ |z=0, (11)

where vτ = (v1, v2) is the tangential part of a 3D vector v , anda ∈ R.




Global existence is assured for weak solutions with constant viscositycoefficients, and only under certain conditions for variable coefficients12.

Strong solutions are only local in time13.

In the constant coefficient case, convergence of weak solutions to astrong solution of the Euler equations has been obtained by Sueur.

12Bresch, Desjardins & Gerard-Varet, J. Math. Pures Appl., 200713Solonnikov, Zap. Naucn. Sem. LOMI Steklov., 1976




Method: a priori estimates in conormal Sobolev spaces, similar to workson the incompressible Navier-Stokes system with Navier BCs (fixedboundary and free surface14).This avoids constructing approximate solutions.

Conormal regularity is necessary to get uniform estimates for boundarylayers. Indeed, we expect the ansatz

(ρ, u)(t, y , z) ∼ (r , v)(t, y , z) +√ε(R,V )(t, y , z/

√ε),

with (r , v) solving the isentropic Euler equation: Lipschitz control at best.

14Masmoudi & Rousset, Arch. Ration. Mech. Anal., 2012 & preprint26 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Method: a priori estimates in conormal Sobolev spaces, similar to workson the incompressible Navier-Stokes system with Navier BCs (fixedboundary and free surface14).This avoids constructing approximate solutions.

Conormal regularity is necessary to get uniform estimates for boundarylayers. Indeed, we expect the ansatz

(ρ, u)(t, y , z) ∼ (r , v)(t, y , z) +√ε(R,V )(t, y , z/

√ε),

with (r , v) solving the isentropic Euler equation: Lipschitz control at best.

14Masmoudi & Rousset, Arch. Ration. Mech. Anal., 2012 & preprint26 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Consider this set of derivatives that are tangent to ∂Ω:

Z0 = ∂t , Z1,2 = ∂y1,2 , Z3 = φ(z)∂z =z

1 + z∂z ,

For m ∈ N and p ∈ 2,∞, we introduce the notations

‖f (t)‖2m,p =

∑α∈N4, |α|≤m

‖(Zαf )(t)‖2Lp(Ω) ,

and |||f |||m,p,T = supt∈[0,T ]

‖f (t)‖m,p

for T ≥ 0.




Consider this set of derivatives that are tangent to ∂Ω:

Z0 = ∂t , Z1,2 = ∂y1,2 , Z3 = φ(z)∂z =z

1 + z∂z ,

For m ∈ N and p ∈ 2,∞, we introduce the notations

‖f (t)‖2m,p =

∑α∈N4, |α|≤m

‖(Zαf )(t)‖2Lp(Ω) ,

and |||f |||m,p,T = supt∈[0,T ]

‖f (t)‖m,p

for T ≥ 0.




The solutions U = (ρ− 1, u) we consider will satisfy

Em(T ,U) = |||U|||2m,2,T + |||∂z uτ |||2m−1,2,T +

∫ T

0

‖∂z (ρ, u3)(s)‖2m−1,2 ds

+ |||∂z uτ |||21,∞,T +

∫ T

0

‖(∂zρ, ∂tzρ)(s)‖21,∞ ds < +∞.

The force term will satisfy, for every T ≥ 0,

Nm(T ,F ) = supt∈[−T ,T ]

(‖F (t)‖2m,2 + ‖∂z F (t)‖2

m−1,2 + ‖F (t)‖22,∞) < +∞

and F (t, x) = 0 for t < 0.




The solutions U = (ρ− 1, u) we consider will satisfy

Em(T ,U) = |||U|||2m,2,T + |||∂z uτ |||2m−1,2,T +

∫ T

0

‖∂z (ρ, u3)(s)‖2m−1,2 ds

+ |||∂z uτ |||21,∞,T +

∫ T

0

‖(∂zρ, ∂tzρ)(s)‖21,∞ ds < +∞.

The force term will satisfy, for every T ≥ 0,

Nm(T ,F ) = supt∈[−T ,T ]

(‖F (t)‖2m,2 + ‖∂z F (t)‖2

m−1,2 + ‖F (t)‖22,∞) < +∞

and F (t, x) = 0 for t < 0.




Theorem 2.1 (submitted, 201415)

Under the conditions stated previously, for ε0 > 0 and m ≥ 7, thereexists T ∗ > 0 such that, for every 0 < ε < ε0, there exists a uniquesolution U to the isentropic Navier-Stokes system (9) with Navierboundary condition (11) in the class of functions satisfyingEm(T ∗,U) < +∞. Moreover, there is uniformly no vacuum on thistime interval.

The family of solutions (Uε)0<ε<ε0 constructed above converges inL2([0,T ∗]× Ω) and in L∞([0,T ∗]× Ω) towards the unique solutionof the isentropic Euler equation ((9) with ε = 0 and thenon-penetration boundary condition) in the same class.

15arxiv 1410.281129 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Remarks.

The results extend to less regular forces or different initialconditions, providing the compatibility conditions yield uniformbounds on Nm(0,U).

Wang and Williams16 proved similar results in the constantcoefficient case by constructing approximate solutions, and needsprior existence of the solution to the Euler system.

16Wang & Williams, Ann. Inst. Fourier, 201230 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Remarks.

The results extend to less regular forces or different initialconditions, providing the compatibility conditions yield uniformbounds on Nm(0,U).

Wang and Williams16 proved similar results in the constantcoefficient case by constructing approximate solutions, and needsprior existence of the solution to the Euler system.

16Wang & Williams, Ann. Inst. Fourier, 201230 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Idea of the proof: a priori estimates + compactness argument.

Theorem 2.2

Under the conditions of Theorem 2.1, for ε0 > 0, there exist T ∗ > 0 anda positive, increasing function Q : R+ → R+ such that, for every0 < ε < ε0 and t ≤ T ∗,

Em(t,U) + |||∂z u3|||21,∞,t + ε

∫ t

0

‖∇u(s)‖2m,2 +

∥∥∇2uτ (s)∥∥2

m,2ds

≤ (t + ε)Q(Em(t,U) +Nm(t,F )).

Throughout, ρ is assumed to be uniformly bounded from below - this ischecked by a bootstrap argument using the fact that, along the flow of u,∂tρ = −ρdiv u, and div u is bounded.





Theorem 2.2


Em(t,U) + |||∂z u3|||21,∞,t + ε

∫ t

0

‖∇u(s)‖2m,2 +

∥∥∇2uτ (s)∥∥2

m,2ds

≤ (t + ε)Q(Em(t,U) +Nm(t,F )).






Theorem 2.2


Em(t,U) + |||∂z u3|||21,∞,t + ε

∫ t

0

‖∇u(s)‖2m,2 +

∥∥∇2uτ (s)∥∥2

m,2ds

≤ (t + ε)Q(Em(t,U) +Nm(t,F )).





The energy estimates on U use the symmetrisable hyperbolic-parabolicstructure of the equation. The system on Uα = ZαU is

(DA0)(ρ)∂tUα+3∑

j=1

(DAj )(U)∂xj Uα−ε(0,div (µ∇u)α+(µ+λ0)∇div uα))

= (0, (ρF + εσ(∇U))α)− Cα,

with the DAj (U) symmetric matrices, σ(∇U) the remainder of theorder-two terms and Cα containing commutators between Zα and theoperators involved in the equation (e.g. [Zα,DAj∂xj ]U).

Basic inequalities are used in the energy estimate.




We now need estimates on ∂z U: these are obtained component bycomponent.

The norms of ∂z u3 are easily controlled by using the equation ofconservation of mass:

ρ∂z u3 = −∂tρ− u · ∇ρ− ρ(∂x u1 + ∂y u2)




We now need estimates on ∂z U: these are obtained component bycomponent.

The norms of ∂z u3 are easily controlled by using the equation ofconservation of mass:

ρ∂z u3 = −∂tρ− u · ∇ρ− ρ(∂x u1 + ∂y u2)




To deal with ∂z uτ , we use a modified vorticity

W = (curl u)τ −2a

µ(ρ)u⊥τ ,

so that we have ρ∂tW + ρu · ∇W − εµ∆W = H and W = 0 on theboundary.

The commutator estimates are standard, except for [Zα, ρu3∂z ]W , inwhich a Hardy-type inequality is used to avoid the appearance of ∇2u.

L∞ bounds are obtained using a variant of the maximum principle, on Win the constant coefficient case, on (curl u)τ otherwise.






µ(ρ)u⊥τ ,









µ(ρ)u⊥τ ,







The mass equation is used to control R = ∂zρ:

lε(∂tR + u · ∇R) + γP(ρ)R = h,

with l = 2µ+ λ0.

The conormal energy estimates are L2 in time.

By following the flow of u, we reduce the equation to the ODElε∂tR + γPR = h, and the Duhamel formula yields the L∞ estimates.




The mass equation is used to control R = ∂zρ:

lε(∂tR + u · ∇R) + γP(ρ)R = h,

with l = 2µ+ λ0.

The conormal energy estimates are L2 in time.

By following the flow of u, we reduce the equation to the ODElε∂tR + γPR = h, and the Duhamel formula yields the L∞ estimates.









A recent preprint obtains a similar result17

using the natural energy structure of the Navier-Stokes system toget estimates on (u,P),

using H1co bounds on ∆P to control |||ρ|||1,∞.

They thus exhibit weaker boundary layers on ρ compared to u, and, onthe domain T× (0, 1), they obtain the rates of convergence:∥∥Uε(t)− UE (t)

∥∥L2 ≤ Cε2 and

∥∥Uε(t)− UE (t)∥∥

L∞ ≤ Cε2/5.

17Wang, Xin & Yong, arxiv 1501.0171837 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



A recent preprint obtains a similar result17

using the natural energy structure of the Navier-Stokes system toget estimates on (u,P),

using H1co bounds on ∆P to control |||ρ|||1,∞.

They thus exhibit weaker boundary layers on ρ compared to u, and, onthe domain T× (0, 1), they obtain the rates of convergence:∥∥Uε(t)− UE (t)

∥∥L2 ≤ Cε2 and

∥∥Uε(t)− UE (t)∥∥

L∞ ≤ Cε2/5.

17Wang, Xin & Yong, arxiv 1501.0171837 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions



Try the same approach as ours on the full Navier-Stokes equation: ∂tρ+ div (ρu) = 0ρ∂tu + ρu · ∇u = div Σ + ρF

ρ∂tθ + ρu · ∇θ + Pdiv u = εκ∆θ + Σ : ∇u + ρu · F .(12)

Internal energy proportional to the temperature θ

Neumann boundary condition for θ

Ideal gas pressure law P(ρ, θ) = kρθ




Disaster strikes! The component-by-component examination of thenormal derivatives has worked for the incompressible and isentropicmodels, but fails in this case!

The problem: the terms involving div u in the equations on ρ and θ.

The field ∂z V = (∂zρ, ∂z u3, ∂zθ) cannot be decoupled.

Work in progress: we can get Hm−1co estimates on ∂z V , but we still need

W 1,∞co bounds: Green function study?




Disaster strikes! The component-by-component examination of thenormal derivatives has worked for the incompressible and isentropicmodels, but fails in this case!

The problem: the terms involving div u in the equations on ρ and θ.

The field ∂z V = (∂zρ, ∂z u3, ∂zθ) cannot be decoupled.

Work in progress: we can get Hm−1co estimates on ∂z V , but we still need

W 1,∞co bounds: Green function study?



Thank you for your attention.


Documents

Boundary layers for Navier-Stokes equations with slip ......Uniform existence and convergence towards Euler Improvements and possible extensions 2 / 40 Matthew Paddick Boundary layers