Boundary Layer problem: Navier-Stokes equations and Euler equations - - PowerPoint PPT Presentation

Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Boundary Layer problem: Navier-Stokes equations and Euler equations

Nikolai Chemetov Joint work with F. Cipriano

CMAF-UL

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

1 Statement of the problem - Vanishing viscosity limit 2 First main result

Estimates, independent of the viscosity Trace value

• f the vorticity

Weak convergence

3 Second main result

Estimates independent of the viscosity Strong convergence

4 Conclusion

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Navier-Stokes equations. Dirichlet condition

           vt + div (v ⊗ v) − ▽p = ν∆v in ΩT := Ω × (0, T), div v = 0 in ΩT := Ω × (0, T), v(x, 0) = v0(x), x ∈ Ω. Homogeneous Dirichlet boundary condition v = 0

• n

ΓT := Γ × (0, T).

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Bucur D., Feireisl E., Necasova S., Boundary Behavior of Viscous Fluids: influence of wall roughness ..., Arch. Rational Mech. Anal., 197, (2010). Priezjev N.V., Troian S.M., Influence of periodic wall roughness on the slip behavior at liquid/solid interfaces: ..., J. Fluid Mech., 554, (2006).

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Navier-slip condition

Flux of the fluid through the boundary Γ v · n = a

• n

ΓT with

• Γ

a dx = 0, ∀t ∈ [0, T], Navier´s slip boundary condition 2D(v)n · s + αv · s = β

• n

ΓT, D(v) = 1

2[∇v + (∇v)T] - the rate-of-strain tensor;

n and s - the external normal and the tangent vector to Γ. α, β describe physical properties of Γ.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Vorticity formulation

The Navier-Stokes equations in terms of the vorticity ω = ∂x1v2 − ∂x2v1 and the velocity v    ∂tω + div (vω) = ν∆ω, divv = 0 in ΩT, ω(x, 0) = ω0(x), x ∈ Ω with ω0 := rot (v0) , v · n = a

• n

ΓT. Navier slip boundary conditions in terms of the vorticity ω = b(v)

• n

ΓT with b(v) := (α − 2k) v · s + b, b = β − 2 a′

s.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

The Euler equations are equivalent to the system            ∂tω + v·∇ω = 0 in ΩT, ω(x, 0) = ω0(x), x ∈ Ω with ω0 := rot (v0) , v · n = a

• n

ΓT. The trajectories for the transport equation start at t = 0 and on the inflow region Γ−, where a < 0. The Navier-slip boundary condition is given just on Γ− ω = b(v).

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Homogeneous case

Navier-Stokes equations with v · n = 0, ω = (α − 2k) v · s

• n

Γ Euler equations with v · n = 0

• n

Γ For α = 2k: Lions P.-L., Math. Topics in Fluid Mechanics (1996) Beir˜ ao da Veiga H., Crispo F., J. Math. Fluid Mech., 13, (2011) For α = 2k: Clopeau T., Mikelic A., Robert R.,Nonlinearity 11, (1998) Lopes Filho M.C., Nussenzveig Lopes H.J., Planas G., SIAM

• Math. Anal., 36, 4, (2005)

Kelliher J., SIAM J. Math. Anal. 38, 1, (2006)

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Non-Homogeneous case

We consider the boundary conditions v · n = a, ω = b(v)

• n

Γ for the Navier-Stokes equations and v · n = a

• n

Γ, ω = b(v)

• n

Γ− for the Euler equations.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Solvability results of the Euler equations

In Lp for any p ∈ (1, ∞] : Chemetov N.V., Antontsev S.N., Physica D: Nonlinear Phenomena, 237, 1, (2008). Chemetov N.V., Cipriano F.; Gavrilyuk, S., Math. Methods

• Appl. Sci. 33, 6, (2010).

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

First main result

Theorem 1 Let a ∈ L∞(0, T; W 1

p (Γ)) ∩ W 1 2 (0, T; W − 1

2

2

(Γ)), b ∈ W 2,1

p (ΓT),

ω0 ∈ W 2

p (Ω),

∀p ∈ (2, +∞]. ∃ a subsequence of {ων, vν}, solutions of the N-S: ων ⇀ ω weakly-* in L∞(0, T; Lp(Ω)), vν → v strongly in C(ΩT),

• ΩT

ω(ψt + v · ∇ψ) dxdt +

• Ω

ω0 ψ(x, 0) dx =

• Γ−

T

a b(v) ψ dxdt for every ψ ∈ C 1,1(ΩT) with ψ = 0 at t = 0 and on Γ+.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Sketch of the proof

1st step : Estimates, independent of the viscosity Lemma ∃ a weak solution {ων, vν} of the Navier-Stokes equations: ||ων||L∞(0,T;Lp(Ω)) C, ||vν||L∞(0,T; W 1

p (Ω)) C,

||∂tvν||L2(ΩT ) C.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

2 important estimates: vν(t)2

L2(Ω) C

• v02

L2(Ω) +

t f (s)ων(s)2

Lp(Ω) ds + 1

• and

ων(t)2

Lp(Ω) C(vν(t)2 L2(Ω) +

t f (s)ων(s)2

Lp(Ω) ds + 1),

where the constants C and f (t) ∈ L1(0, T) are independent of ν.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

2nd step : Trace value of the vorticity Let B be an extension in ΩT of the initial condition B

• t=0 = ω0

and boundary condition B

• ΓT = b(vν) ;

d(x)-the distance function from x ∈ Ω to Γ. Lemma lim

σ→0+

• lim

ν→0+

1 σ

• ΩT ∩[0<d<σ]

|ων − B| | (vν ▽ d) |− dxdt

• = 0.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

3d step : Weak convergence Taking ν → 0: ων ⇀ ω weakly-* in L∞(0, T; Lp(Ω)), vν → v strongly in C(ΩT), we will obtain the Euler equation

• ΩT

ω(ψt + v · ∇ψ) dxdt +

• Ω

ω0 ψ(x, 0) dx =

• Γ−

T

a b(v) ψ dxdt

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Second main result

Theorem 2 Let ω0 ∈ Lp(Ω), a ∈ L∞(0, T; W 1

p (Γ)),

b ∈ L2(0, T; W 1

p (Γ) ∩ W 1 2 (0, T; W − 1

p

p

(Γ)), ∀p ∈ (2, +∞]. ∃ a subsequence of {ων, vν}, solutions of N-S: ων → ω strongly in Lr(0, T; Lp(Ω)), ∀r < ∞, vν → v strongly in Lr(0, T; W 1

p (Ω)),

• ΩT

β(ω)

• ψt+ v·∇ψ
• dxdt+
• Ω

β(ω0) ψ(x, 0) dx =

• Γ−

T

aβ(b(v))ψ dxdt holds for any β ∈ C(R) and any test function ψ.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Sketch of the proof

Uniform estimates Lemma ∃ a weak solution {ων, vν} of N-S: ||ων||L∞(0,T;Lp(Ω))

• C,

||vν||L∞(0,T; W 1

p (Ω))

• C,

||∂t(vν − ∇ A)||L2(ΩT ) C, where A is the solution of the system −∆A = 0 in Ω,

∂A ∂n = a

• n

Γ

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Strong convergence Malek J., Necas J., Rokyta M., Ruzicka M., Weak and measure-valued solutions to evolutionary PDEs. (1996). We denote h(x) = min{δ, d(x)} for any x ∈ Ω with a fixed δ > 0. We consider the cut-off function ξν(x) := 1 − exp(−M + νL ν h(x)), ∀ν > 0, with vνL∞(ΩT ) M,

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Multiplying the Navier-Stokes equations by η′(ων)ξνψ, with η ∈ C 2(R) convex and ψ ∈ C ∞

0 (R2+1) non-negative, we obtain

• ΩT

η(ων) [ψt + vν · ∇ψ + ν ∆ψ] ξν + 2 ν η(ων) (∇ψ · ∇ξν) dxdt +

• Ω

η(ω0) ψ(x, 0) ξν dx −

• Ω

η(ω(x, T)) ψ(x, T) ξν dx +

• ΓT

(M + νL)η(b) ψ dxdt0

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

vν → v strongly in L∞(0, T; L2(Ω)), |ων − c|+ ⇀ z1, |ων − c|− ⇀ z2, (ων − c) ⇀ ω−c = z1−z2, weakly-* in L∞(0, T; Lp(Ω)),

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Hence

• ΩT

z1 ψt + v · ∇ψ

• dx dt

+M

• ΓT

|b − c|+ψ dx dt +

• Ω

|ω0 − c|+ψ(x, 0) dx0, and

• ΩT

z2 ψt + v · ∇ψ

• dx dt

+M

• ΓT

|b − c|−ψ dx dt +

• Ω

|ω0 − c|−ψ(x, 0) dx0. The strong convergence of ων to ω follows from the result: If z1z2 = 0 in ΩT, then z1 = |ω − c|+, z2 = |ω − c|−.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Hence |ων−c|± ⇀ |ω−c|± weakly-* in L∞(0, T; Lp(Ω)), ∀c ∈ R, that implies the strong convergence ων → ω in Lr(0, T; Lp(Ω)), vν → v in Lr(0, T; W 1

p (Ω)), ∀r,

and the limit {ω, v} satisfies

• ΩT

|ω − c|

• ψt + v · ∇ψ
• dx dt

+M

• ΓT

|b − c| ψ dx dt +

• Ω

|ω0 − c| ψ(0, x) dx 0.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Renormalization property Applying Boyer F., Trace theorems ..., Differential and integral equations, 18, 8 (2005) we derive

• ΩT

β(ω)

• ψt + v · ∇ψ
• dx dt +
• Ω

β(ω0) ψ(x, 0) dx =

• ΓT

aβ(b(v))ψ dx dt for any β ∈ C(R) and arbitrary ψ ∈ C 1,1(ΩT) with ψ = 0 at t = T.

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Outline Statement of the problem - Vanishing viscosity limit First main result Second main result Conclusion

Conclusion

• The sharp change of the velocity inside the boundary layer could

be avoided if the mathematical model allows fluid motion along the boundary.

• By this reason the vanishing viscosity convergence results were
• btained due to Navier slip type conditions.

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