[PPT] - The global well-posedness for the compressible viscous fluid flow in PowerPoint Presentation

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The global well-posedness for the compressible viscous fluid flow in 3D exterior domains

Yoshihiro Shibata

Math. Department and RISE, Waseda University

Mathflows 2015, Porquerolles

Sept. 13–18, 2015.

Partially supported by Top Global University Project and JSPS Grant-in-aid for Scientific Research (S) # 24224004 Joint work with Yuko Enomoto (Shibaura Institute of Technology).

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. . (NP)        ρt + div (ρu) = 0 in Ω × (0, T), ρ(ut + u · ∇u) − µ∆u − ν∇div u + ∇P(ρ) = 0 in Ω × (0, T), u|∂Ω = 0, (ρ, u)|t=0 = (ρ∗ + ρ0, u0) in Ω. Ω ⊂ R3 : exterior domain (R3\Ω is bounded) ∂Ω : boundary of Ω, sufficiently smooth ρ∗ > 0 : mass density of the reference body Ω µ > 0, ν > 0 : viscosity coefficients P(ρ) : C∞ function of ρ > 0 ρ = ρ(x, t) : density u = (u1(x, t), u2(x, t), u3(x, t)) : velocity .

Assumption

. . . . . . . . P ′(ρ) > 0 for any ρ > 0 and P(ρ∗) = 0 If not, we consider P(ρ) − P(ρ∗) instead of P(ρ).

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History (1/3)

Local well-posedness

Cauchy Problem

J.Nash (1962) N.Itaya (1970) A.I.Volpert-S.I.Hudjaev (1972)

I.B.V.P.

V.A.Solonnikov (1976),(1981)

Global well-posedness

A.Matsumura & T.Nishida (1980), (1981)

Ω : R3 or exterior domain, Assumption: ∥(ρ0, u0)∥H3 ≤ ε ρ − ρ∗ ∈ C0([0, ∞), H3(Ω)) ∩ C1([0, ∞), H2(Ω)), u ∈ C0([0, ∞), H3(Ω)) ∩ C1([0, ∞), H1(Ω)), ρt, ∇ρ, ut ∈ L2((0, ∞), H2(Ω)), ∇u ∈ L2((0, ∞), H3(Ω))

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History (2/3)

. .

Global well-posedness

A.Valli (1983), A.Valli & W.Zajaczkowski (1986)

non-homogenious boundaly domain, periodic solution

V.A.Solonnikov (1995)

Ω : bounded domain, u ∈ W ℓ+2,ℓ/2+1

2

(ℓ > 1/2)

M.Kawashita (2002)

Ω = R3, Cauchy problem, Assumption : ∥(ρ0, u0)∥H2 ≤ ε (ρ, u) ∈ C0([0, ∞), H2), ∇u ∈ L2((0, ∞), H2), ∇ρ ∈ L2((0, ∞), H1)

R.Danchin (2000)

Critical space for the Cauchy problem

Y.Kagei & T.Kobayashi (2002), (2005)

Half space

Y.Kagei & S.Kawashima (2006), Kagei (2012)

Layer domain

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History (3/3)

Lp- Lq type decay

Cauchy problem G.Ponce (1985), Y.Wang & Z.Tan (2011) Exterior domain K.Deckelnick (1992), T.Kobayashi & S. (1999), Y.Enomoto & S (2012)

Lp-Lq maximal regularity

Y.Enomoto & S (2013) Local well-posedness in general unbounded domain Global well-posedness in a bounded domain u ∈ Lp((0, T), W 2

q (Ω)) ∩ W 1 q ((0, T), Lq(Ω)),

ρ ∈ W 1

p ((0, T), W 1 q (Ω))

(Lagrangean).

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Comments on the local well-posedness

.

Using the Lagrange transformation to change the transport

equation: ρt + u · ∇ρ to ∂tρ = ⇒ Quasilinear parabolic system.

We prove the maximal Lp in time Lq in space maximal regularity

for the linearized equation. ρt + ρ∗div u = f, ρ∗ut − α∆u − β∇div u + γ∗∇ρ = g, u|Γ = 0, (ρ, u)|t=0 = (ρ0, u0).

To prove the maximal regularity, we construct an R bounded

solution operator Aλ to the corresponding generalized resolvent problem. λρ + ρ∗div u = ˆ f, ρ∗λu − α∆u − β∇div u + γ∗∇ρ = ˆ g, u|Γ = 0.

Apply the Weis operator valued Fourier multipler theorem to the

representation formula: u(·, t) = L−1

λ [AλL[f, g](λ)](·, t) with the

help of Laplace transform L and its inverse transform L−1 in time variable t and its co-variable λ.

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Global well-posedness in the bounded domain case

.

exponential stability of the analytic semigroup associated with the

linearized equations.

spectral analysis to the resolvent problem:

λρ+ρ∗div u = f, ρ∗λu−α∆u−β∇div u+γ∗∇ρ = g in Ω, u|Γ = 0.

λ ̸= 0 =

⇒ ρ = λ−1(f − ρ∗div u) ρ∗λu − α∆u − (β + γ∗ρ∗λ−1)∇div u = g′ in Ω, u|Γ = 0.

New prob. ρ∗µu − α∆u − (β + γ∗ρ∗λ−1)∇div u = g′ in Ω, u|Γ = 0.
uniqueness implies the unique existence, thus

ρ∗λu − α∆u − (β + γ∗ρ∗λ−1)∇div u = 0 in Ω, u|Γ = 0 = ⇒ u = 0

λ = 0 case: We have to prove the unique existence theorem of the

problem: div u = f, µ∆u + β∇div u − γ∇ρ = g in Ω, u|Γ = 0, Thus, ∫

Ω f dx =

∫

Ω div u dx =

∫

Γ u · n dσ = 0 is necessary to show

the exponential decay.

To prove the global well-posedness for the small data, we assume

that ∫

Ω ρ0 dx = 0.

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Global well-posedness

. . Ω ⊂ R3 : exterior domain, (ρ0, u0) ∈ H2, ∥(ρ0, u0)∥H2 ≤ ε ≪ 1 compatibility condition : u0|∂Ω = 0 = ⇒ the problem (NP) admits a unique solution (ρ, u) ρ ∈ C0([0, ∞), H2) ∩ C1([0, ∞), H1), u ∈ C0([0, ∞), H2) ∩ C1([0, ∞), L2) ∇ρ, ρt ∈ L2((0, ∞), H1), ut ∈ L2((0, ∞), H1), ∇u ∈ L2((0, ∞), H2) Moreover, ∥(ρ0, u0)∥H2 + ∥(ρ0, u0)∥L1 = δ ≪ 1 = ⇒ ∥(ρ(·, t), u(·, t))∥L2 ≤ Ct− 3

4 δ,

∥∇(ρ(·, t), u(·, t))∥H1 ≤ Ct− 5

4 δ as t → ∞ Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow

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Linearized equation

. . (LP)        ρt + ρ∗div u = 0 in Ω × (0, ∞), ut − µ∗∆u − ν∗∇div u + γ∗∇ρ = 0 in Ω × (0, ∞), u|∂Ω = 0, (ρ, u)|t=0 = (ρ0, u0) in Ω where Ω ⊂ RN (N ≥ 3) is an exterior domain.

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Lp-Lq decay for linearized equation

. . Let 1 < q < ∞ and let N ≥ 2. Then problem (LP) generates C0 semigroup {T(t)}t≥0 on Hq(Ω) = {(ρ0, u0) ∈ W 1

q (Ω) × Lq(Ω)} which is

analytic. Let 1 ≤ q ≤ 2 ≤ p ≤ ∞, let N ≥ 2, and let [ρ0, u0]p,q = ∥ρ0∥W 1

p + ∥u0∥Lp + ∥(ρ0, u0)∥Lq. Then,

∥(ρ, u)(·, t)∥Lp ≤ Cp,qt− N

2

(

1 q − 1 p

)

[ρ, u0]p,q, ∥∇(ρ, u)(·, t)∥Lp ≤ Cp,q    t− N

2

(

1 q − 1 p

) − 1

2 [ρ, u0]p,q,

p ≤ N t− N

2q [ρ, u0]p,q,

p ≥ N ∥∇2u(·, t)∥Lp ≤ Cp,q    t− N

2

(

1 q − 1 p

) −1[ρ, u0]p,q,

p ≤ N/2, t− N

2q [ρ, u0]p,q.

p ≥ N/2

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Idea of proof

.

. Global well-posedness : A.Matsumura & T.Nishida method

first energy is obtained directly in Ω
To obtain higher order energy estimate, we consider the half-space

problem: x3 > 0.

energy method for tangential derivative and time derivative
To estimate normal derivative D3, we use the formula

D3(ρt + u · ∇ρ) + δ∗D3ρ = (· · · there are no D2

3 terms).

To estimate D2

3ρ, we use the formula

D2

3(ρt + u · ∇ρ) + δ∗D2 3ρ = · · ·

Multiplying this formula by D2

3ρ, we have

1 2 d dt∥D2

3ρ(·, t)∥2 + δ∗∥D2 3ρ(·, t)∥2 − (div uD2 3ρ, D2 3ρ) = · · ·

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The worst term of the nonlinear estimate: ∫ t ∥∇2ρ∇u∥2

L2 ds ≤

∫ t ∥∇2ρ∥2

L2∥∇u∥2 L∞ ds

≤ C ∫ t ∥∇2ρ∥2

L2∥∇u∥2 H2 ds

≤ C sup

0<s<t

∥∇2ρ(·, s)∥2

L2

∫ t ∥∇u∥2

H2 ds

To estimate nonlinear terms, we use ∥v∥L6 ≤ ∥∇v∥L2, ∥v∥L2(Ω∩BR) ≤ CR∥∇v∥L2, ∥v∥L∞ ≤ C∥v∥L6 ≤ C∥v∥H1 In this way, we can enclosed our estimations in (ρ, u) ∈ L∞((0, ∞), H2(Ω)), (ρt, ut) ∈ L∞((0, ∞), L2(Ω)) ∇u ∈ L2((0, ∞), H2(Ω)), ut, ∇ρ, ρt ∈ L2((0, ∞), H1(Ω))

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Lagrange approach in a unbounded domain

. .

x = ξ +

∫ t

0 u(ξ, s) ds, u being the velocity field in the Lagrange

coordinate:

x = ξ +

∫ ∞ u(ξ, s) ds − ∫ ∞

t

u(·, s) ds

the equation in the Euler coordinate =

⇒ θt + ρ∗div v + ρ∗div (V1(v)v) + θdiv (v + V1(v)v) = 0, ρ∗vt − µ∆v − µ′∇div v + P ′(ρ∗)∇θ + θvt + V2(v)∇2v + V4(v) ∫ t ∇2v ds∇v + (P ′(ρ∗ + θ) − P ′(ρ∗))∇θ + P ′(ρ∗ + θ)V3(v)∇θ = ⃗ 0,

Vi(v) = Wi(

∫ t

0 ∇v(ξ, s) ds) (i = 1, 2, 3, 4) where Wi are suitable

polynomials and Wi(0) = 0 (i = 1, 2, 3).

Vi(v) = Wi(

∫ ∞ ∇v(·, s) ds) + ∫ ∞

t

∇v dsW ′

i(·).

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Free boundary problem in the exterior domain; Incompressible case

. .

Ω ⊂ RN (N ≥ 3): exterior domain
Ωt: time evolution of Ω
Γ: boundary of Ω, Γt: boundary of Ωt.

(1)        ∂tv + (v · ∇v) − Div (νD(v) − πI) = 0, div v = 0 in Ωt, (νD(v) − πI)nt = 0, Vt = nt · v

n Γt,

Ω0 = Ω, v|t=0 = v0, in Ω.

ν > 0 viscousity constant.

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New equations

. .

φ ∈ C∞

0 (RN): φ = 1 near Γ.

u : the velocity field in the Lagrange coordinate
x = ξ + φ(ξ)

∫ t

0 u(ξ, s) ds (Modified Lagrange transformation).

x = ξ + φ(ξ)

∫ ∞ u(ξ, s) ds − φ(ξ) ∫ ∞

t

u(ξ, s) ds (2)            ∂tu − A2(ξ, ∂x)(u, π) = F(u), in Ω, div {(I + B0(ξ))u} = div D0(u) = D1(u) in Ω, C1(ξ, ∂x)(u, π)n = G(u)

n Γ,

v|t=0 = v0, in Ω.

F(u) : ∇2u

∫ ∞ ∇(φu) ds, ∇u ∫ ∞ ∇2(φu) ds, u · ∇u.

D0(u) : u

∫ ∞ ∇(φu) ds, D(u) : ∇u ∫ ∞ ∇(φu) ds

G(u) : ∇u

∫ ∞ ∇(φu) ds

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Thoerem

.

. Let N ≥ 3 and let q1 and q2 be exponents such that N < q2 < ∞ and 1/q1 = 1/q2 + 1/N and q1 > 2. Let b, p and p′ = p/(p − 1) be numbers satisfying the conditions: N q1 > b > 1 p′ , (N q1 − b ) p > 1, ( b − N 2q2 ) p > 1, b > N 2q1 , ( N 2q2 + 1 2 ) p′ < 1, bp′ > 1, ( b − N 2q2 ) p′ > 1, N q2 + 2 p < 1. (3) Then, there exists an ϵ > 0 such that ∥u0∥B2(1−1/p)

q2,p

+ ∥u0∥Lq1/2 ≤ ϵ, then problem (2) admits a unique solution u with u ∈ Lp((0, ∞), W 2

q2(Ω)N) ∩ W 1 p ((0, ∞), Lq2(Ω)N)

possessing the estimate: ∫ T ((1 + t)b∥u(·, s)∥W 1

∞(Ω))p ds

+ ∫ T ((1 + s)(b− N

2q1 )∥u(·, s)∥W 1 q1(Ω))p ds + ( sup

0<s<T

(1 + s)

N 2q1 ∥u(·, s)∥q1)p

+ ∫ T ((1 + s)(b− N

2q2 )(∥u(·, s)∥W 2 q2(Ω) + ∥∂tu(·, s)∥Lq2(Ω)))p ds ≤ Mϵ.

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We consider the equations:

ut − Au = f, Bu|Γ = g, u|t=0 = 0

time -shifted equations have the exponential stability:

vt + λ0v − Av = f Bv|Γ = g, v|t=0 = 0

u = v + w, so that

wt − Aw = −λ0v Bw|Γ = 0, w|t=0 = 0.

w =

∫ t

0 T(t − s)f(s) ds with f(s) = −λ0v(s).

Lp-Lq decay estimate =

⇒ ∥∇iw(t)∥∞ ≤ C ∫ t ∥∇iT(t − s)f(s)∥∞ ds ≤ C ∫ t−1 (t − s)− N

2 ( 2 q1 − 1 ∞)+ i 2 ∥f(s)∥q1/2 ds

+ C ∫ t

t−1

(t − s)−( N

2q2 + i 2 +ϵ)∥f(s)∥q2 ds

(i = 0, 1).

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.

here (N q1 + i 2 − b ) p ≥ (N q1 − b ) p > 1. thus ∫ T

2

(tbv0(t))p dt ≤ CM.

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.

v1(t) = ∫ t−1

t/2

(t − s)

− (

N q1 + i 2

)

∥f(·, s)∥q1/2 ds ≤ (∫ t−1

t/2

(t − s)

− (

N q1 + i 2

)

ds )1/p′ × (∫ t−1

t/2

(t − s)

− (

N q1 + i 2

)

∥f(·, s)∥p

q1/2 ds

)1/p . thus ∫ T (v1(t)tb)p dt ≤ (N q1 + i 2 − 1 )−p/p′ ∫ T

2

(∫ t−1

t/2

(t − s)

− (

N q1 + i 2

)

(sb∥f(·, s)∥q1/2)p ds ) dt change the integration order ≤ (N q1 + i 2 − 1 )−p/p′ ∫ T −1

1

(sb∥f(·, s)∥q1/2)p(∫ 2s

s+1

(t − s)

− (

N q1 + i 2

)

dt ) ds ≤ (N q1 + i 2 − 1 )−p M

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1

((1 + s)b∥f(·, s)∥q2)p(∫ s+1

s

(t − s)−

(

N 2q2 + i 2 +ϵ

))

dt ) ds ≤ ( 1 − N 2q2 − i 2 − ϵ )−p ∫ T

1

((1 + s)b∥f(·, s)∥q2)p ds ≤ ( 1 − N 2q2 − i 2 − ϵ )−p M.

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. . Let ∫ T ((1 + s)b∥f(·, s)∥q1/2)p ds + ∫ T ((1 + s)b∥f(·, s)∥q2)p ds + ∫ T ((1 + s)

N 2q1 ∥f(·, s)∥q3)p ds = M < ∞

(1/q3 = 1/q1 + 1/q2). Then, ∫ T ((1 + t)b∥u(·, s)∥W 1

∞)p ds +

∫ T ((1 + s)(b− N

2q1 )∥u(·, s)∥W 1 q1)p ds

+ ( sup

0<t<T

(1 + t)

N 2q1 ∥u(·, t)∥W 1 q1)p

+ ∫ T {(1 + s)(b− N

2q2 )(∥u(·, s)∥W 2 q2 + ∥∂tu(·, s)∥Lq2)}p ds ≤ CM Y.Shibata (Waseda Univ. ) Compressible viscous fluid flow

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