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Maximal Regularity for the Initial-Boundary Value Problem of some Evolution Equations

Yoshihiro Shibata

Waseda University

VIII Workshop in Partial Differential Equations

August 25-28, Rio 2009

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 1 / 23

Analytic Semigroup

X : Banach space with norm ∥ · ∥ A : D(A)(⊂ X) → X : densely defined closed operator {T(t)}t≥0 is an analytic semigroup associated with A ⇐⇒

T(t) is a bounded linear operator on X such that T(t)T(s) = T(t + s) for t, s > 0 T(t) is strongly continuous in t > 0 limt→0+ ∥T(t)x − x∥ = 0 for any x ∈ X (*) Ax = limt→0+ T(t)x − x t for x ∈ D(A)(= {x ∈ X | the limit (*) exists}) T(t) is extended as an analytic function of t to a sector {t ∈ C | | arg t| < θ0} with some θ0 ∈ (0, π/2). T(t)x ∈ D(A), d

dtT(t)a = AT(t)a (t > 0) for any x ∈ X

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 2 / 23

Generation of analytic semigroup

Given 0 < γ < π/2 we set

Gγ = {λ ∈ C\{0} | | arg λ| < π 2+γ} Assumption ∃ b ∈ R > 0 and 0 < γ < π/2 such that for any λ ∈ b + Gγ the resolvent (λ − A)−1 exists and it holds that ∥(λ − A)−1x∥ ≤ M|λ|−1∥x∥

for any x ∈ X. The operator A generates an analytic semigroup {T(t)}t≥0 :

∥T(t)x∥ ≤ Mebt∥x∥, ∥ d dtT(t)x∥ ≤ Mt−1ebt∥x∥

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 3 / 23

Lp maximal regularity

Let us consider the abstract Cauchy problem: (**)

u′ − Au = f (0 < t < T), u(0) = u0

Solution class: W1

p((0, T), X) ∩ Lp((0, T), D(A))} = Mp((0, T), A).

Initial data class: Zp(A)(X, D(A))1−(1/p),p = the trace class of Mp((0, T), A)

A has an Lp maximal regularity ⇐⇒

For any f ∈ Lp((0, T), X) and u0 ∈ Zp(A), the problem (**) admits a unique solution u(t) ∈ W1

p((0, T), X) ∩ Lp((0, T), D(A))

Closed graph theorem of S. Banach =⇒

∥u∥W1

p((0,T),X) + ∥Au∥Lp((0,T),X) ≤ CT(∥f∥Lp((0,T),X) + ∥u0∥Zp(A)).

A has an Lp maximal regularity =⇒ A generates an analytic

semigroup.

⇐= ? (H. Br´

ezis) : The answer is No in general (Kalton and Lancien).

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 4 / 23

An example of non-linear evolution eq. of parabolic type

vt − ∆v + F(v) = 0 (t > 0), v(0) = v0 F(v) : semi-linear ⇐⇒ F(v) is a non-linear function of v and ∇v. F(v) : quasi-linear ⇐⇒ F(v) = G(v)∇2v, G(v) is a semi-linear function.

Semi-linear =⇒ Analytic semigroup approach.

{T(t)}t≥0 : analytic semigroup generated by −∆. The Duhamel principle =⇒ v(t) = T(t)v0 − ∫ t T(t − s)F(s) ds

One of the tools to solve this equation is to use the Lq-Lr estimate:

∥T(t)f∥Lr ≤ Cq,rMebtt− k

2− n 2

( 1

q− 1 r

)

∥f∥Lq, (1 < q ≤ r < ∞) ∥∇T(t)f∥Lr ≤ Cq,rMebtt− 1

2− n 2

( 1

q− 1 r

)

∥f∥Lq (1 < q ≤ r ≤ n)

(1)

∥∇kT(t)f∥Lq ≤ Mebtt− k

2 ∥f∥Lq (k = 0, 1, 2) PLUS ∥v∥Lr ≤ C∥v∥

n ( 1

q − 1 r

) W1

q

∥v∥

1−n ( 1

q− 1 r

) Lq

(n

( 1

q − 1 r

)

)PLUS semigroup property =⇒ (1)

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 5 / 23

Quasi-linear =⇒ we need Lp-Lq maximal regularity approach.

∥∇2v(t)∥Lq ≤ ∥∇2T(t)v0∥Lq + ∫ t

0 ∥∇2T(t − s)F(s)∥Lq ds

∥∇2T(t − s)F(s)∥Lq ≤ C(t − s)−1∥G(s)∇v(s)∥Lq

This approach fails because the singurality: (t − s)−1 appears.

Lp-Lq maximal regularity theorem is applied to the linearized eq. vt − ∆v = −G(w)∇2w =⇒ ∥vt∥Lp((0,T),Lq) + ∥v∥Lp((0,T),W2

q)

≤ CT(∥v0∥(Lq,W2

q)1−(1/p),p + ∥G(w)∇2w∥Lp((0,T),Lq)

toghether with the embedding theorem:

W1

p((0, T), Lq)) ∩ Lp((0, T), W2 q) ⊂ BC([0, T], (Lq, W2 q)1−(1/p),p)

implies at least a local in time unique existence theorem

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 6 / 23

Navier-Stokes equation

vt + (v · ∇)v − µ∆v + ∇p = f, div v = 0 v = (v1, . . . , vn) velocity vector, p scalor pressure v · ∇θ = ∑n

j=1 vjDjθ, vt + (v · ∇)v = Dv

Dt : material derivative µ = 1/R, R = ρLV/ˆ µ : Reynolds number. ρ : mass, L: length, V: velocity, ˆ µ: viscosity coefficient

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 7 / 23

Fujita-Kato principle

The scaling: vλ(x, y) = λ2v(λx, λ2t), pλ(x, t) = λ2p(λx, λ2t) does not change the NS eq.

∥vλ∥Lp((0,∞),Lq(Rn)) = ∥v∥Lp((0,∞),Lq(Rn)) =⇒ 2 p + n q = 1 (Serrin Condition) Theorem (Kato)

(2)

vt + (v · ∇)v − µ∆v + ∇p = 0, div v = 0 in Rn × (0, ∞), v|t=0 = a

Given initial data a ∈ Ln(Rn) with div a = 0, there exists a time T > 0 such that the Navier-Stokes equation (2) admits a unique strong solution

u ∈ C0([0, T), Ln(Rn)) ∩ C0((0, T), Lq(Rn) ∩ W1

n(Rn)) with some q ∈ (n, ∞).

Moreover, if ∥a∥Ln(Rn) is small enough, then T = ∞.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 8 / 23

Observation for the global in time existence theorem

If the local existence time T depends only on ∥a∥Ln

=⇒

the scaling argument implies the prolongation of a local in time solution to any time interval

=⇒

global in time unique existence of the NS equation for any initial data in Ln !! But, unfortunately T depends on some properties of ∥∇u(·, t)∥Ln and

∥u(·, t)∥Lq, so that to get global in time unique existence theorem, so far we

have to assume some smallness on the initial data.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 9 / 23

Navier-Stokes equation in the time dependent domain

Ω(t)

vt + v · ∇v − µ∆v + ∇p = 0, div v = 0 (x ∈ Ω(t), t > 0), v|t=0 = v0

with some boundary condition. Free boundary problem A viscous incompressible fluid flow past rotating bodies.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 10 / 23

Reduction to the Quasi-linear equation

Suitable change of variables and unknown function:

Free boundary problem ⇐⇒ Lagrangian coordinate Rotating obstacle ⇐⇒ Galdi transform, Inoue-Wakimoto transform

implies the quasi-linear equation:

vt − µ∆v + ∇p + F(t, v, ∇v, ∇2v, ∇p) = 0 (x ∈ Ω, t > 0), div (v + G(t, v)) = 0 (x ∈ Ω, t > 0), vt=0 = v0

with suitable boundary condition on a fixed domain Ω.

F and G are some nonlinear functions such that F|t=0 = G|t=0 = 0.

Serrin cond. =⇒ Lp((0, T), Lq(Ω)) type maximal regularity is necessary.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 11 / 23

Some results about Lp maximal regularity

• 1. Stokes Equation

ut − µ∆u + ∇p = f, div u = 0 (x ∈ Ω, t > 0) u|t=0 = u0

with the following boundary condition on the boundary Γ: Non-slip: u = 0 (Solonnikov, Giga-Sohr) Slip: D(u)ν− < ν, D(u)ν > ν = g, u · ν = 0 (Saal, Shimada) Free bc: µD(u)ν − pν = g (Shibata-Shimizu, Solonnikov).

ν stands for the unit outer normal to Γ and < ·, · > is the standard

inner-product in Rn, D(u) = ∇u + TD(u) (the symmetric part of ∇u).

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 12 / 23

Stokes equation with height function (Shibata-Shimizu, Solonnikov, Pr¨ uß- Simonet)(a linearized prob of the NS eq. with surface tension)

ut − µ∆u + ∇p = f, div u = 0 (x ∈ Ω, t > 0) ηt − ν · u = η0 (x ∈ Γ, t > 0) D(u)ν− < ν, D(u)ν > ν = h′ (x ∈ Γ, t > 0) < ν, S(u, p)ν > +σ∆Γη = hn (x ∈ Γ, t > 0) u|t=0 = u0 ∆Γ stands for the Laplace-Beltrami operator on Γ, σ > 0.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 13 / 23

Examples of domain Ω

Bounded domain, Half-space Rn

+ = {x ∈ Rn | xn > 0}

Layer L = {x ∈ Rn | 0 < xn < 1} Tube {x ∈ Rn | x′ ∈ D, xn ∈ R} with D being a bounded domain in Rn−1, Exterior domain : Ω ∩ BR = BR = {x ∈ Rn | |x| > R} (for some R >> 1). Perturbed half-space: Ω ∩ BR = Rn

+ ∩ BR (for some R >> 1).

Perturbed layer: Ω ∩ BR = L ∩ BR (for some R >> 1). Apperture domain: Ω ∩ BR = ({x ∈ Rn | xn < 0} ∪ {x ∈ Rn | xn > 1}) ∩ BR (for some R >> 1).

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 14 / 23

Results

initial data u0 belongs to the time trace class the right member f ∈ Lp((0, T), Lq(Ω)), 1 < p, q < ∞. Non-slip: u ∈ W1

p((0, T), Lq(Ω)) ∩ Lp((0, T), W2 q(Ω)).

Slip: g ∈ H

1 2

p ((0, T), Lq(Ω)) ∩ Lp((0, T), W1 q(Ω)) =⇒

u ∈ W1

p((0, T), Lq(Ω)) ∩ Lp((0, T), W2 q(Ω))

Free bc: g ∈ H

1 2

p ((0, T), Lq(Ω)) ∩ Lp((0, T), W1 q(Ω)) =⇒

u ∈ W1

p((0, T), Lq(Ω)) ∩ Lp((0, T), W2 q(Ω))

Stokes with height function: g ∈ H

1 2

p ((0, T), Lq(Ω)) ∩ Lp((0, T), W1 q(Ω)),

η0 ∈ Lp((0, T), W2

q(Ω)) =⇒ u ∈ W1 p((0, T), Lq(Ω)) ∩ Lp((0, T), W2 q(Ω)),

η ∈ W1

p((0, T), W 2− 1

q

q

(Γ)) ∩ Lp((0, T), W

3− 1

q

q

(Γ)).

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 15 / 23

• 2. Thermoelastic plate equation

utt + ∆2u + ∆θ = f, θt − ∆θ − ∆ut = g (x ∈ Ω, t > 0) (u, θ)|t=0 = (u0, θ0)

with the following boundar conditions: hinged bc: u = ∆u = θ = 0 (Munˆ

• s-Rivera & Racke)

Dirichlet bc: u = ∂νu = θ = 0

(x ∈ Γ, t > 0) (Y. Naito)

free bc: ∆u − (1 − β)∆Γu + θ = 0, ∂ν(∆u + (1 − β)∆Γu) = 0, ∂νθ = 0

(0 < β ≤ 1) (R. Denk-Y. Enomoto-Y. Shibata) Theorem Ω is a bounded domain in Rn (n ≥ 2). Let initial data u0 belong to the time

trace class and the right member f , g ∈ Lp((0, T), Lq(Ω)) ( 1 < p, q < ∞). Then the initial boundary value problem for the thermoelastic plate equation adimits a unique solutions

u ∈ Lp((0, T), W4

q(Ω)) ∩ W1 p((0, T), W2 q(Ω)) ∩ W2 p((0, T), Lq(Ω))

θ ∈ W1

p((0, T), Lq(Ω)) ∩ Lp((0, T), W2 q(Ω)).

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 16 / 23

Technical Tool

Fourier multiplier theorem Let 1 < p < ∞. Let m(ξ) (ξ ∈ Rn) be a function defined on Rn \ {0} such that

|Dα

ξ m(ξ)| ≤ Cα|ξ|−|α|

(|α| ≤ n).

Define the operator M f by the formula: M f = F −1[m(ξ) ˆ

f (ξ)] Then, M

is a bounded linear operator on Lp(Rn) possessing the estimate:

∥ M f ∥Lp(Rn) ≤ Cp,n(max

|α|≤n Cα)∥ f ∥Lp(Rn).

Operator valued Fourier multiplie theorem (L. Weis) Let X and Y be UMD Banach spaces and 1 < p < ∞. Assume that

M ∈ C1(R \ {0}, L(X, Y)) and that { M(τ) | τ ∈ R \ {0}} and {τM′(τ) | τ ∈ R \ {0}} are R bounded with bound D. Define the

• perator TM f by the formula: TM f = F −1[M(τ) ˆ

f(τ)]. Then, TM is a

bounded linear operator from L(R, X) into L(R, Y) and

∥TM f ∥Lp(R,Y) ≤ CD∥ f∥Lp(R,X)

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 17 / 23

R boundedness and some reduction

Let A ⊂ L(X, Y). We call A is R-bounded if there exists p ∈ [1, ∞) and C > 0 such that

∑

δj∈{−1,1}

|

m

∑

j=1

δjTj aj|p ≤ C ∑

δj∈{−1,1}

|

m

∑

j=1

δjaj|p

for any aj ∈ X, Tj ∈ A (j = 1, . . . , m) and any naural number m. The infinimum of such C is called the R bounde of A. A sufficient condition: Let A ⊂ L(Lq(Ω), Lq(Ω)). A is R bounded if and only if there exists an M > 0 such that

∥ ( m ∑

j=1

|Tj fj |2) 1

2 ∥Lq(Ω) ≤ M∥

( m ∑

j=1

| fj |2) 1

2 ∥Lq(Ω)

for any T1, . . . , Tm ∈ A, f1, . . . , fm ∈ Lq(Ω) and any natural number m.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 18 / 23

Stragegy of our proof of the generation of analytic semigroup and Lp-Lq maximal regularity

To show the R boundedness of the set {λ(λ − A)−1 | λ ∈ Σθ0} for the resolvent problem: (λ − A)u = f in Rn and Rn

+.

To show the resolvent estimate and Lp-Lq maximal regularity theorem in the bent-half space Hω.

Hω = {x ∈ Rn | xn > ω(x′) (x′ = (x1, . . . , xn−1) ∈ Rn−1} ∥∇′ω∥L∞ << 1.

Using a suitable partition of unity, we show that the unique existence

• f solutions to the resolvent problem for large λ in Ω. ⇐⇒ The

generation of an analytic semigroup in Ω. Localizing the problem, we show the Lp-Lq maximal regularity estimate like:

∥ut∥Lp((0,T),Lq(Ω)) + ∥u∥Lp((0,T),W2

q(Ω)) ≤ C(∥f∥Lp((0,T),Lq(Ω)) + ∥u∥Lp((0,T),W1 q(Ω)))

with zero initial condition.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 19 / 23

An explanation of our approach to the resolvent problem and Lp-Lq maximal regularity

Initial-boundary value problem:

ut − ∆u = 0 (x ∈ Rn

+, t > 0), Dnu|xn=0 = G|xn=0, u|t=0 = 0.

Laplace transform =⇒

λvλ − ∆vλ = 0 (x ∈ Rn

+), Dnvλ|xn=0 = gλ|xn=0

u(·, t) = L−1[vλ](t) = 1

2πeγt ∫ R eiτtvγ+iτ dτ

Fourier transform with respect to x′ = (x1, . . . , xn−1):

(λ + |ξ′|2 − D2

n)ˆ

vλ(ξ′, xn) = 0 xn > 0, Dnˆ vλ(ξ′, 0) = ˆ gλ(ξ′, 0).

solution formula:

vλ(x) = F −1

ξ′

[e−√

λ+|ξ′|2xn

√ λ + |ξ′|2 ˆ gλ(ξ′, 0) ] (x′).

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 20 / 23

Volevich trick:

vλ(x) = ∫ ∞ ∂ ∂yn F −1

ξ′

[e−√

λ+|ξ′|2(xn+yn)

√ λ + |ξ′|2 ˆ gλ(ξ′, yn) ] (x′) dyn = ∫ ∞ F −1

ξ′

[e−√

λ+|ξ′|2(xn+yn)

√ λ + |ξ′|2 ˆ H(λ, ξ′, yn) ] (x′) dyn ˆ H(λ, ξ′, yn) = √ λ + |ξ′|2ˆ gλ(ξ′, yn) − Dnˆ gλ(ξ′, yn).

Define the operator Aλ by the formula:

AλG = λvλ = ∫ ∞ F −1

ξ′

[λe−√

λ+|ξ′|2(xn+yn)

√ λ + |ξ′|2 ˆ H(λ, ξ′, yn) ] (x′) dyn

Define: kλ(x) = F −1

ξ′

[λe−√

λ+|ξ′|2xn

√

λ+|ξ′|2

] (x′) λvλ = AλG = ∫

Rn

+ kλ(x′ − y′, xn + yn)H(λ, y) dy. Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 21 / 23

Lemma(Denk-Hieber-Pr¨ uß): Let 1 < q < ∞ and Ω be a domain in Rn. Let Lλ (λ ∈ Λ) be a kernel operator defined by the formula:

Lλf = ∫

Ω

ℓλ(x, y)f(y) dy.

Assume that there exists a function ℓ(x, y) such that |ℓλ(x, y)| ≤ ℓ(x, y) for any (x, y) ∈ Ω × Ω. Define the kernel operator L by the formula:

Lf = ∫

Ω ℓ(x, y)f(y) dy. If there exists a constant CL such that

∥Lf∥Lq(Ω) ≤ CL∥f∥Lq(Ω), then the family of the operators {Lλ | λ ∈ Λ} is R-bounded and its R-bound ≤ CL. |kλ(x)| ≤ Cθ0|x|−n whenever λ ∈ Σθ0 (θ0 ∈ (π/2, π)).

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 22 / 23

Set Kf =

∫ Cθ0f(y) (|x′ − y′|2 + (xn + yn)2)

n 2 . Then, we see that

∥Kf∥Lq(Rn

+) ≤ Cn,qCθ0∥f∥Lq(Rn +)

(f ∈ Lq(Rn)).

Denk-Hieber-Pr¨ uß lemma =⇒ the family of the operator {Aλ | λ ∈ Σθ0}

• n Lq(Rn

+) is R bounded and its R bound is less than Cn,qCθ0.

In particular, we have the estimate:

|λ|∥vλ∥Lq(Rn

+) ≤ C(|λ| 1 2 ∥gλ∥Lq(Rn) + ∥∇gλ∥Lq(Rn +))

Lp-Lq maximal regularity estimate: ∥ut∥Lp(R,Lq(Rn)) ≤ C(∥G∥

H

1 2 p (R,Lq(Rn +)) + ∥∇G∥Lp(R,Lq(Rn +))).

In particular, we can get the Lp-Lq maximal regularity theorem as well as the resolvent estimate at the same time.

Yoshihiro Shibata (Waseda Univ.) Maximal Regularity for the initial-boundary value problem of some evolution equations of parabolic August 25-28, Rio 2009 23 / 23