Navier-Stokes equations with time-dependent boundary conditions - - PowerPoint PPT Presentation

Navier-Stokes equations

with time-dependent boundary conditions

Sylvie Monniaux

in collaboration with El Maati Ouhabaz (IMB, Bordeaux - France) I2M, Université Aix-Marseille - France

Mathflows – September 2015, Porquerolles

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The equations

Given τ > 0, on Ω ⊂ R3 a bounded C 1,1 or convex domain, we consider the following initial condition problem: (RNS)                  ∂tu − ∆u + ∇π − u × curl u = in [0, τ] × Ω div u = in [0, τ] × Ω ν · u =

• n

[0, τ] × ∂Ω ν × curl u = βu

• n

[0, τ] × ∂Ω u(0) = u0 in Ω. where ν is the unit exterior normal, β : [0, τ] × ∂Ω → S3(R) is bounded, nonnegative a.e. and admits ν as eigenvector a.e.

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About the nonlinearity

The identity for u smooth vector field (u · ∇)u = −u × curl u + 1 2∇|u|2 allows to rewrite the usual transport term in the Navier-Stokes equations, so that ∂tu − ∆u + ∇p + (u · ∇)u = 0 becomes ∂tu − ∆u + ∇π − u × curl u = 0 with π = p + 1

2|u|2.

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About the boundary conditions

The boundary conditions split into two parts: ν · u = 0 on ∂Ω non penetration; ν × curl u = βu Robin-type condition (with friction coefficient β). Remark For smooth Ω, the above boundary conditions are equivalent to ν · u = 0 on ∂Ω,

• S(u, π)ν
• tan + (β − 2W)u = 0 on ∂Ω

Navier’s slip boundary condition with friction, where W is the Weingarten map on ∂Ω (depends on the geometry of Ω), and S(u, π) is the Cauchy stress tensor: S(u, π) := 1 2 (∇u + (∇u)⊤) − πI3.

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The Leray projection

Define H :=

• u ∈ L2(Ω, R3); div u = 0 in Ω and ν · u = 0 on ∂Ω
• .

The following orthogonal direct sum holds: L2(Ω, R3) = H

⊥

⊕ ∇H1(Ω). The projection P : L2(Ω, R3) → H is the classical Leray projection. Define now WT :=

• u ∈ L2, div u ∈ L2, curl u ∈ L2, ν · u = 0 on ∂Ω
• and V := WT ∩ H. Then

P : WT − → V. Remark (Amrouche, Bernardi, Dauge, Girault [M2AS, 1998]) In the case of C 1,1 or convex bounded domains, WT ֒ → H1(Ω, R3).

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The Leray projection

Define H :=

• u ∈ L2(Ω, R3); div u = 0 in Ω and ν · u = 0 on ∂Ω
• .

The following orthogonal direct sum holds: L2(Ω, R3) = H

⊥

⊕ ∇H1(Ω). The projection P : L2(Ω, R3) → H is the classical Leray projection. Define now WT :=

• u ∈ L2, div u ∈ L2, curl u ∈ L2, ν · u = 0 on ∂Ω
• and V := WT ∩ H. Then

P : WT − → V. Remark (Amrouche, Bernardi, Dauge, Girault [M2AS, 1998]) In the case of C 1,1 or convex bounded domains, WT ֒ → H1(Ω, R3).

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The result

Theorem (Monniaux-Ouhabaz, 2015) ∃ ε > 0 s.t. ∀ u0 ∈ V1/2 := [H, V]1/2 with u0V1/2 ≤ ε, there is a unique u ∈ H1(0, τ; V′

1/2) with t → Aβ(t)u(t) ∈ L2(0, τ; V′ 1/2) and π ∈ L2(0, τ; H1/2)

such that (u, π) satisfies (RNS) in the sense of distributions. Recall (RNS)      ∂tu − ∆u + ∇π − u × curl u = in [0, τ] × Ω div u = in [0, τ] × Ω ν · u = 0, ν × curl u = βu

• n

[0, τ] × ∂Ω.

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The bilinear form

On the Hilbert space H, we consider the bilinear form aβ : V × V − → R aβ(u, v) := curl u, curl vΩ + βu, v∂Ω. Thanks to the properties of β, aβ is symmetric, closed and coercive. We denote by Aβ the associated (self adjoint) operator D(Aβ) =

• u ∈ V; ∃ v ∈ H : aβ(u, φ) = v, φΩ
• ,

Aβu = v.

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The operator

Theorem (Monniaux-Ouhabaz, 2015) The operator Aβ is given by D(Aβ) =

• u ∈ L2(Ω, R3); div u = 0 in Ω, curl u ∈ L2(Ω, R3),

curl curl u ∈ L2(Ω, R3), ν · u = 0 and ν × curl u = βu on ∂Ω

• ,

Aβu = P(curl curl u) = −∆u + ∇p, u ∈ D(Aβ), p ∈ H1(Ω). In addition, −Aβ generates an analytic semigroup of contractions on H and D(A1/2

β ) = V.

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Non autonomous maximal regularity

Under the above assumptions on β: β : [0, τ] × ∂Ω → S3(R) bounded measurable, piecewise Hölder continuous in t of order α > 1/4, admits a.e. ν as eigenvector for a.e. t ∈ (0, τ) and β(t, x)ξ · ξ ≥ 0 for almost all (t, x) ∈ [0, τ] × ∂Ω and all ξ ∈ R3, we have that ∗ ∀ u0 ∈ H ∀ f ∈ L2(0, τ; V′) ∃! u ∈ H1(0, τ, V′) ∩ L2(0, τ; V)

• r

(Lions, 1957) ∗ ∀ u0 ∈ V ∀ f ∈ L2(0, τ; H) ∃! u ∈ H1(0, τ, H) with Aβu ∈ L2(0, τ; H) (Arendt-Monniaux, 2015) such that ∂tu + Aβu = f and u(0) = u0, with the accompanying estimates.

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Non autonomous maximal regularity, cont’d

Interpolation spaces Define V1/2 = [H, V]1/2 and V′

1/2 = [V′, H]1/2:

D(Aβ)

Aβ

• A1/2

β

• V

Aβ

• A1/2

β

• V1/2

A1/2

β

• H

A1/2

β

• V′

1/2

V′

Under the previous assumptions on β, interpolating between the two results, ∀ u0 ∈ V1/2, ∀ f ∈ L2(0, τ; V′

1/2)

∃! u ∈ H1(0, τ; V′

1/2) with Aβu ∈ L2(0, τ; V′ 1/2)

such that ∂tu + Aβu = f and u(0) = u0, with the accompanying estimates.

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Non autonomous maximal regularity, cont’d

Interpolation spaces Define V1/2 = [H, V]1/2 and V′

1/2 = [V′, H]1/2:

D(Aβ)

Aβ

• A1/2

β

• V

Aβ

• A1/2

β

• V1/2

A1/2

β

• H

A1/2

β

• V′

1/2

V′

Under the previous assumptions on β, interpolating between the two results, ∀ u0 ∈ V1/2, ∀ f ∈ L2(0, τ; V′

1/2)

∃! u ∈ H1(0, τ; V′

1/2) with Aβu ∈ L2(0, τ; V′ 1/2)

such that ∂tu + Aβu = f and u(0) = u0, with the accompanying estimates.

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The space of maximal regularity

E :=

• u ∈ H1(0, τ; V′

1/2); Aβu ∈ L2(0, τ; V′ 1/2) and u(0) ∈ V1/2

• Lemma

E ֒ → L4(0, τ; V). Idea of the proof. Aβu ∈ L2(0, τ; V′

1/2) implies A1/2 β u ∈ L2(0, τ; V1/2);

u ∈ H1(0, τ; V′

1/2) and Aβu ∈ L2(0, τ; V′ 1/2) imply, by interpolation, that

A1/2

β u ∈ H1/2(0, τ; V′ 1/2);

we conclude by interpolation that A1/2

β u ∈ H1/4(0, τ; H) ֒

→ L4(0, τ; H).

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Fixed point theorem

The idea is to rewrite the time-dependent Robin-Navier-Stokes system as a fixed point problem u = a + B(u, u), u ∈ E, with a the solution of ∂ta + Aβa = 0, a(0) = u0 ∈ V1/2, and w = B(u, u) the solution of ∂tw + Aβw = P(u × curl u) ∈ L2(0, τ; V′

1/2),

w(0) = 0. Lemma (Picard’s contraction principle) Assume that aE ≤

1 4BE×E→E . Then v → a + B(v, v) has a unique fixed

point in E.

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Fixed point theorem

The idea is to rewrite the time-dependent Robin-Navier-Stokes system as a fixed point problem u = a + B(u, u), u ∈ E, with a the solution of ∂ta + Aβa = 0, a(0) = u0 ∈ V1/2, and w = B(u, u) the solution of ∂tw + Aβw = P(u × curl u) ∈ L2(0, τ; V′

1/2),

w(0) = 0. Lemma (Picard’s contraction principle) Assume that aE ≤

1 4BE×E→E . Then v → a + B(v, v) has a unique fixed

point in E.

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Proof of: P(u × curl u) ∈ L2(0, τ; V′

1/2) for u ∈ E

If u ∈ E, then curl u ∈ L4(0, τ, L2(Ω, R3)); u ∈ L4(0, τ; V) and V ֒ → H1(Ω, R3) ֒ → L6(Ω, R3), so that u ∈ L4(0, τ, L6(Ω, R3)). This proves that u × curl u ∈ L2(0, τ, L3/2(Ω, R3)) since 1

6 + 1 2 = 2 3 and 1 4 + 1 4 = 1 2, and therefore

P(u × curl u) ∈ L2(0, τ, V′

1/2)

since V1/2 ֒ → L3 (and then PL3/2 ֒ → V′

1/2).

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Thank you for your attention!

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