Emmrich E., Wittbold P. (editors) Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series

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Maximal regularity and applications to PDEs 265

for z ∈ Σ

= {w ∈ C \ {0}; |arg(w )| < π − θ} with θ ∈ [0,

[, t = |z|

−

, x ∈ Ω,

k ∈ N and (h(k))

k≥1

satisfying

h(k) ≤ c

(k + 1)

−δ

for some δ >

Remark 3.7. A semigroup satisfying the Gaussian estimates (3.2) satisﬁes the two

bounds above for all 1 < q

≤ q

< ∞.

These kinds of bounds for q

= 2 are easier to prove than the pointwise Gaussian

estimates (3.2), in particular, the second one (3.6). Indeed, it is, for instance for diver-

gence form elliptic operators, only a matter of partial integration, as we will see below

for the Navier–Lamé operator (see Theorem 3.9).

Theorem 3.8 (Kunstmann, 2008). Let Ω ⊂ R

be a domain. Let A be the negative

generator of an analytic semigroup (T (t))

t≥0

in L

(Ω) satisfying (3.6) with q

> q

2. Then A has the maximal L

-regularity property in L

(Ω) for all q ∈ [2, q

References for the proof. This result is due to P. C. Kunstmann [24]. The original state-

ment uses (3.5) instead of (3.6), and is presented in a more general context: instead of

Ω ⊂ R

, Ω is assumed to be a space of homogeneous type.

In the following, we study the Navier–Lamé operator

L = −µ∆ −(λ + µ)∇div , µ > 0, λ + µ ≥ 0,

supplemented by homogeneous Dirichlet boundary conditions, which appears in linear

elasticity theory. To be precise, let L be the operator in L

(Ω; R

) associated with the

sesquilinear form

`(u, v) = µ

Ω

∇u ·

∇v dx + (λ + µ)

Ω

div u div v dx, u, v ∈ H

(Ω; R

Since ` is symmetric, continuous and coercive, the operator L is self-adjoint and gen-

erates a bounded analytic semigroup in L

(Ω; R

Theorem 3.9. Let Ω be a Lipschitz domain in R

(n ≥ 3). Then the Navier–Lamé

operator with Dirichlet boundary conditions has the maximal L

-regularity property

in L

(Ω) for all q ∈]

n+2

n−2

Proof. Fix an arbitrary point x ∈ Ω,

z ∈ Σ

= {w ∈ C \ {0}; |arg(w)| < π − θ},

t = |z|

−

and an arbitrary partition of unity {η

; j ∈ N} of R

such that

∈ C

∞

(B(x, 2t); R), η

∈ C

∞

B(x, 2

j+1

t) \B(x, 2

j−1

t); R

0 ≤ η

≤ 1, |∇η

| ≤

j−1

∞

j=0

= 1,

(3.7)

266 Sylvie Monniaux

where B(x, r) is the ball in R

with center at x ∈ R

and radius r > 0, decompose

f ∈ L

(Ω, R

) as

f =

∞

j=0

, f

= η

f and set u =

∞

j=0

, u

= (zI + L)

−1

∈ D(L). (3.8)

We will prove that for all p ∈ [2,

n−2

], there exist two constants C, c > 0 such that

|z|

Ω∩B(x,t)

≤ Ce

−c2

−

)

Ω

∀j ∈ N. (3.9)

This will be done in three steps.

Step 1. Pick a new family of functions (ξ

)

j≥1

such that ξ

∈ C

∞

(B(x, 2

j−1

t); R).

Taking the L

-pairing of ξ

with both sides of zu

+ Lu

= f

, and keeping in mind

that ξ

= 0 for each j ≥ 1, we may write, based on integration by parts,

Ω

dy + µ

Ω

|∇u

dy + (λ + µ)

Ω

|div u

Ω

|∇ξ

||u

||ξ

µ|∇u

| + (λ + µ)|div u

i´

dy.

(3.10)

From this, via the Cauchy–Schwarz inequality and a standard trick that allows us to

absorb similar terms with small coefﬁcients in the left-hand side, we get

|z|

Ω

dy ≤ C

Ω

|∇ξ

dy (3.11)

and, since λ + µ ≥ 0,

Ω

|∇u

dy ≤ C

Ω

|∇ξ

|∇u

dy. (3.12)

Step 2. Similarly as in [6], we now replace the cutoff function ξ

in (3.11) by another

cutoff function e

− 1 (which has the same properties as ξ

), with

|z|

√

Ck∇ξ

∞

, j ≥ 2.

In a ﬁrst stage, this yields

Ω

− 1|

dy ≤

Ω

dy,

then further

Ω

dy ≤ 4

Ω

dy, (3.13)

Maximal regularity and applications to PDEs 267

in view of the generic, elementary observation

kf − gk ≤

kfk implies kfk ≤ 2kgk.

If we now assume that the original cutoff functions (ξ

)

j≥2

also satisfy

0 ≤ ξ

≤ 1, ξ

= 1 on B(x, t) and k∇ξ

∞

≤

it follows from the deﬁnition of α

that α

≥ c2

and from (3.13) that

Ω∩B(x,t)

dy ≤ 4

Ω

dy ≤

|z|

Ω

dy,

the second inequality coming from the fact that −L generates an analytic semigroup.

This gives then

|z|

Ω∩B(x,t)

dy ≤ Ce

−c2

Ω

dy. (3.14)

The same procedure allows to estimate ∇u on B(x, t) using (3.12) as follows

|z|

Ω∩B(x,t)

|∇u

dy ≤ Ce

−c2

Ω

dy. (3.15)

Those two estimates are also valid if j = 0 since the resolvent of L is bounded in

(Ω; R

Step 3. Let p = 2

∗

n−2

and D ⊂ R

(n ≥ 3) be a Lipschitz domain of diameter

R > 0. The continuous embedding of H

(D) into L

(D) then reads, after rescaling,

−

)

|u|

≤ C

h³

|u|

+ R

|∇u|

. (3.16)

Combining this inequality with (3.14) and (3.15), and keeping in mind that |z| =

we have for all j ∈ N

|z|

Ω∩B(x,t)

n−2

≤ C t

−1

−c2

Ω

. (3.17)

By interpolation, using (3.14) and (3.17), the generalized Gaussian bound (3.9) is

proved for all 2 ≤ p ≤

n−2

By Theorem 3.8, we may conclude that L has the maximal L

-regularity property

in the space L

(Ω; R

) for all q ∈ [2,

n−2

[. By duality (since L is self-adjoint), we

can prove that L has the maximal L

-regularity property in L

(Ω; R

) also for all

q ∈]

n+2

, 2], which proves Theorem 3.9.

268 Sylvie Monniaux

4 Applications to partial differential equations

In this section, we will apply maximal L

-regularity results to show the uniqueness of

solutions of certain partial differential equations. We start with a toy problem, studied

by F. Weissler [42]. This will show the way to prove uniqueness of mild solutions of

the incompressible Navier–Stokes system.

4.1 Existence and uniqueness for a semilinear heat equation

We are interested in the initial-boundary value problem

∂u

∂t

− ∆u = u

in ]0, T [×Ω,

u(t, x) = 0 on ]0, T [×∂Ω,

u(0) = u

in Ω,

(4.1)

where T > 0 and Ω ⊂ R

is a domain with no particular regularity at the boundary.

We assume that n ≥ 4. The critical space where we are looking for solutions is L

(Ω)

with p =

. This space is critical in the sense that if p >

, then the nonlinearity u

is a “small” perturbation of the linear part −∆u and if p <

, then the nonlinearity

“wins” and the methods applied here are not appropriate. Our purpose is to show that

a solution u ∈ C ([0, T ]; L

(Ω)) of (4.1) (in an integral sense deﬁned below) is unique

in the space C ([0, T ]; L

(Ω)).

Deﬁnition 4.1. A function u is said to be an integral solution of (4.1) with u

∈ L

(Ω)

on [0, τ] if u ∈ C ([0, τ]; L

(Ω)) and if u satisﬁes

u(t) = T (t)u

T (t −s)(u(s)

) ds, t ∈ [0, τ ],

where (T (t))

t≥0

is the semigroup generated by the Dirichlet–Laplacian in L

(Ω),

which we denote by −A.

We ﬁrst study the question of existence.

Theorem 4.2 (F. Weissler, 1981). Let n ≥ 4. For any initial value u

∈ L

(Ω), there

exists τ ∈]0, T ] and an integral solution u ∈ C ([0, τ ]; L

(Ω)) of (4.1) in [0, τ ]. If

is sufﬁciently small, then τ = T .

Proof. We will show the local existence of an integral solution of (4.1) via a ﬁxed

point method. We reformulate the problem as to ﬁnd a Banach space E

containing

C ([0, T ]; L

(Ω)) such that a = T (·)u

∈ E

and there exists u ∈ E

verifying

u = a + B(u, u),

where B is the bilinear operator deﬁned by

B(u, v)(t) =

T (t −s)(u(s)v(s)) ds, t ∈ [0, T ].

Maximal regularity and applications to PDEs 269

We need B to be continuous on E

× E

. We choose

u ∈ C ([0, T ]; L

(Ω)); t 7→ t

u(t) ∈ C ([0, T ]; L

(Ω))

and we deﬁne the norm in this space to be

kuk

= sup

0<t<T

ku(t)k

+ sup

0<t<T

ku(t)k

u ∈ E

This space E

endowed with its norm is a Banach space. We remark ﬁrst that (T (t))

t≥0

is a bounded analytic semigroup in L

(Ω) for all p ∈]1, ∞[, which implies, in particu-

lar, that for all α ≥ 0, there exists a constant c

p,α

> 0 such that

T (t)k

L (L

(Ω))

≤ c

p,α

, t > 0.

Therefore, it is easy to check that a ∈ E

. We will show next that

B : E

× E

→ E

is continuous. Let u, v ∈ E

. Then we have

t 7→ t

u(t)v(t) ∈ C ([0, T ]; L

(Ω))

with norm bounded by kuk

kvk

. Therefore, we obtain

−

(u(t)v(t))k

≤ ckuk

kvk

for all t ∈ [0, T ] because of the continuous embedding W

(Ω) ⊂ L

(Ω). We then

have

B(u, v)(t) =

T (t −s)(u(s)v(s)) ds

√

t −s A

T (t −s)A

−

(

√

s u(s)v(s))

√

t −s

√

which gives, for all t ∈ [0, T ], the estimate

kB(u, v)(t)k

≤ c c

kuk

kvk

√

t −s

√

≤ πc c

kuk

kvk

since

√

t −s

√

ds =

√

1 −σ

√

dσ =

−

√

1 −4r

dr = π.

The same arguments are used to estimate t

kB(u, v)(t)k

. For u, v ∈ E

, we have

−

(u(t)v(t))k

≤ ckuk

kvk

270 Sylvie Monniaux

because of t

−

(u(t)v(t)) ∈ L

(Ω) and the continuous embedding W

(Ω) ⊂

(Ω) in dimension n. Therefore, we have

B(u, v)(t)k

≤ c c

kuk

kvk

(t −s)

−

≤ c c

kuk

kvk

(1 −σ)

−

dσ

We can ﬁnish the proof of existence by applying the Picard ﬁxed point theorem (see

Theorem A.1 below) as long as kak

≤

4kBk

. This is the case if ku

is small

enough. The argument must be adapted a little if ku

is not small by adjusting T so

that the result remains true.

We now turn to the question of uniqueness.

Theorem 4.3 (F. Weissler, 1981). Assume that n ≥ 5. Let u

, u

∈ C ([0, T ]; L

(Ω))

be two integral solutions of (4.1) for the same initial value u

∈ L

(Ω) on some

interval [0, T ] of local existence. Then u

= u

on [0, T ].

The proof of the foregoing theorem will be prepared by the following result.

Lemma 4.4. The bilinear operator

B : L

(0, T ; L

(Ω)) ×C ([0, T ]; L

(Ω)) → L

(0, T ; L

(Ω))

is bounded for all q ∈]1, ∞[.

Proof. For u ∈ L

(0, T ; L

(Ω)) and v ∈ C ([0, T ]; L

(Ω)) the product uv is in

(0, T ; L

(Ω)). Therefore,

f : t 7→ A

−1

(u(t)v(t)) ∈ L

(0, T ; L

(Ω)).

Since the Dirichlet–Laplacian enjoys the maximal L

-regularity in L

(Ω) (see Propo-

sition 3.2), we have

kB(u, v)k

(0,T ;L

(Ω))

t 7→ A

AT (t −s)f(s) ds

(0,T ;L

(Ω))

≤ Ckuk

(0,T ;L

(Ω))

kvk

C ([0,T ];L

(Ω))

the constant C coming from the maximal L

-regularity property of A in L

(Ω).

Proof of Theorem 4.3. With the same notations as in the previous proof, u

and u

are

both solutions of the equation

u = a + B(u, u).

If we denote by v the difference between u

and u

, then v must satisfy the equation

v = B(v, u

+ u

Maximal regularity and applications to PDEs 271

with u

, u

, v ∈ C ([0, T ]; L

(Ω)) and u

(0) = u

, v(0) = 0. To prove that

v = 0 on a small interval [0, τ] (which implies then that v = 0 on the whole interval

[0, T ]), we need the following auxiliary lemma, which proof lies below, and this is

where we use the maximal regularity property for the Dirichlet–Laplacian in L

(Ω).

Let ε > 0 be ﬁxed. Choose u

0,ε

∈ C

∞

(Ω) such that

− u

0,ε

< ε.

We decompose B(v, u

+ u

) into three parts:

B(v, u

+ u

) = B(v, u

+ u

− 2u

) + 2B(v, u

− u

0,ε

) + 2B(v, u

0,ε

We can estimate the ﬁrst two parts thanks to Lemma 4.4. This gives then for any

τ ∈]0, T ]

kB(v, u

+ u

− 2u

(0,τ;L

(Ω))

≤ Ckvk

(0,τ;L

(Ω))

− u

C ([0,τ];L

(Ω))

+ ku

− u

C ([0,τ];L

(Ω))

(4.2)

and

kB(v, u

− u

0,ε

(0,τ;L

(Ω))

≤ Cεkvk

(0,τ;L

(Ω))

. (4.3)

Since ku

− u

C ([0,τ];L

(Ω))

→ 0 as τ → 0 for i = 1, 2, it remains to estimate the last

part B(v, u

0,ε

). Since u

0,ε

∈ C

∞

(Ω), we have vu

∈ L

(0, τ; L

(Ω)) and

kB(v, u

0,ε

)(t)k

≤ M t

1−

kvk

(0,τ;L

(Ω))

0,ε

∞

It is now obvious that kB(v, u

0,ε

(0,τ;L

(Ω))

→ 0 as τ → 0. Combining this last

result with the estimates (4.2) and (4.3), we obtain that for ε and τ small enough,

kB(v, u

+ u

(0,τ;L

(Ω))

≤

kvk

(0,τ;L

(Ω))

Since v is solution of v = B(v, u

+ u

) on [0, T ], and then in particular on [0, τ ],

this implies that v = 0 almost everywhere on [0, τ ]. Since moreover v is continuous

on [0, τ ], we conclude that v = 0 (everywhere) on [0, τ]. These arguments show that

{t ∈ [0, T ]; v = 0} is an open set in [0, T ] and by continuity of v, this set is also

closed. Since it is not empty, it is necessarily equal to the whole interval [0, T ] by

connectedness.

4.2 Uniqueness for the incompressible Navier–Stokes system

The incompressible Navier–Stokes equations in the whole space R

reads as

∂u

∂t

− ∆u + ∇π + (u · ∇)u = 0 in ]0, T [×R

div u = 0 in ]0, T [×R

u(0) = u

in R

(4.4)

272 Sylvie Monniaux

where

u : [0, T ] ×R

→ R

and π : [0, T ] ×R

→ R

denote the velocity of a ﬂuid and its pressure. The notation (u ·∇)v for u and v vector

ﬁelds stands for

i=1

∂

v. We assume that there is no external force. Throughout

this section, we also assume that n ≥ 3.

We can reformulate this system in a functional analytic setting as follows: Let A be

the operator with domain W

2,p

; R

) (for a p ∈]1, ∞[), given by

Au = P(−∆u) where P = I + ∇(−∆)

−1

div,

P is the Leray projection and is bounded in L

; R

) since the Riesz projections are

bounded. As in the case of the semilinear heat equation above, there is a critical space

for (4.4). In the scale of Lebesgue spaces, the critical space here is L

; R

), which

means that if p > n, then the nonlinearity (u ·∇)u appears as a “small” perturbation of

the linear part P(−∆u) and if p < n, then the nonlinearity “wins”.

The existence of integral solutions of (4.4) (see Deﬁnition 4.5 below) for an initial

condition u

∈ L

; R

) has been proved by T. Kato in 1984 in [23]. The proof

is similar to the proof of the existence of solutions for the semilinear heat equation

(Theorem 4.2). A good reference for this problem is the book by P. G. Lemarié–

Rieusset [29].

Deﬁnition 4.5. A function u is said to be an integral solution of (4.4) on [0, τ ] for

∈ L

; R

) with div u

= 0 if u ∈ C ([0, τ ]; L

; R

)) and if u satisﬁes

u(t) = e

t∆

− P

(t−s)∆

∇ ·(u(s) ⊗ u(s)) ds, t ∈ [0, τ],

where (e

t∆

)

t≥0

is the semigroup generated by the Laplacian ∆.

Remark 4.6. (i) Since u is a divergence-free vector ﬁeld, we have

(u ·∇)u =

i=1

∂

u =

i=1

∂

u) = ∇ ·(u ⊗ u),

which explains the form of u in Deﬁnition 4.5.

(ii) In the previous deﬁnition, we want the semigroup (e

t∆

)

t≥0

to act on a distribution

∇(u ⊗ u) ∈ W

−1,

. This makes sense since in the case of the whole space R

the heat semigroup acts on all W

s,p

; R

), s ∈ R, p ∈]1, ∞[.

Theorem 4.7 (T. Kato, 1984). Let u

∈ L

; R

) satisfy divu

= 0. Then there

exists T > 0 and an integral solution u ∈ C ([0, T ]; L

; R

) of (4.4). If ku

small enough, then the solution u is global (i.e., we can take T = ∞).

Idea of the proof. The proof follows the line of the proof of Theorem 4.2 by working

on the space

u ∈ C ([0, T ]; L

; R

); div u = 0 in R

and

t 7→

√

t∇u(t) ∈ C ([0, T ]; L

; R

n×n

)

Maximal regularity and applications to PDEs 273

endowed with the norm

kuk

= sup

0<t<T

ku(t)k

+ sup

0<t<T

√

tk∇u(t)k

We are looking for u ∈ E

solution of u = a + B(u, u) where B is deﬁned by

B(u, v)(t) = −P

(t−s)∆

∇·(u(s) ⊗v(s) +v(s) ⊗u(s))

ds, t ∈ [0, T ], (4.5)

and a(t) = e

t∆

, t ∈ [0, T ].

Theorem 4.8 (G. Furioli, P. G. Lemarié–Rieusset, E. Terraneo, 2000). Let u

, u

two integral solutions of (4.4) for the same initial value u

∈ L

; R

) satisfying

div u

= 0. Then u

= u

on [0, T ].

Proof. This result was ﬁrst proved by G. Furioli, P. G. Lemarié–Rieusset and E. Ter-

raneo in [18] (see also the very nice review on the subject by M. Cannone [11]). The

proof presented here is based on [33] and of the same spirit as the proof of Theo-

rem 4.3. The basic idea is to reformulate the problem of uniqueness as to show that

u = u

−u

is equal to zero on the interval [0, T ]. The function u satisﬁes the equation

u = B(u, u

+ u

) where B is deﬁned by (4.5) above. As shown by F. Oru [35], the

bilinear operator

B : C ([0, T ]; L

; R

)) ×C ([0, T ]; L

; R

)) → C ([0, T ]; L

; R

))

is not bounded. Had it been continuous, the proof of uniqueness of an integral solution

of the Navier–Stokes system would have been straightforward. The idea of [18], in

dimension n = 3, was then to lower the regularity of the space L

; R

) and to

consider a Besov space E instead (or, as shown by Y. Meyer in [31], the weak L

space, namely L

3,∞

; R

)) to obtain a bounded bilinear operator B in C ([0, T ]; E)×

C ([0, T ]; E).

The proof of [33] relies on a slightly different idea: instead of weaken the regularity

of the space in the x-variable, we consider a Lebesgue space L

in time instead of the

space of continuous functions in time. As in the proof of Theorem 4.3, we write

u = B(u, u

− u

) + B(u, u

− u

) + 2B(u, u

− u

0,ε

) + 2B(u, u

0,ε

where u

0,ε

is chosen in C

∞

; R

) close to u

in the L

-norm. To be able to show

that u = 0 on a small interval [0, τ ] (τ > 0) with the same method as in the proof of

Theorem 4.3, we only need a result of the same kind as Lemma 4.4, see Lemma 4.9

below. At that point, the argument goes exactly as before.

Lemma 4.9. The bilinear operator

B : L

(0, T ; L

; R

)) ×C ([0, T ]; L

; R

)) → L

(0, T ; L

; R

))

is bounded for all p ∈]1, ∞[. More precisely, for all p ∈]1, ∞[, there exists a constant

> 0 such that for all u ∈ L

(0, T ; L

; R

)) and all v ∈ C ([0, T ]; L

; R

)),

we have

kB(u, v)k

(0,T ;L

))

≤ c

kuk

(0,T ;L

))

kvk

C ([0,T ];L

))

274 Sylvie Monniaux

Proof. To prove this lemma is exactly where the maximal L

-regularity property of the

Laplacian comes in. We rewrite B(u, v) as

B(u, v)(t) = P(−∆)

−(t−s)(−∆)

(−∆)

−1

f(s) ds,

where f = −

∇ · (u ⊗ v + v ⊗ u). What we only have to prove then is that we

can estimate (−∆)

−1

f in the L

(0, T ; L

; R

))-norm with respect to the norm of

u in L

(0, T ; L

; R

)) and the norm of v in C ([0, T ]; L

; R

)). Indeed, since

we know that −∆ has the maximal L

-regularity property, this would imply the result.

Employing the continuous embedding W

; R

) ⊂ L

; R

), we ﬁnd

k(−∆)

−1

(0,T ;L

))

≤ Ck(−∆)

−1

(0,T ;W

))

≤ Ckfk

(0,T ;W

−1,

))

≤ Ck(u ⊗v + v ⊗ u)k

(0,T ;L

))

≤ Ckuk

)

kvk

C ([0,T ];L

))

and therefore we have proved Lemma 4.9.

5 Maximal regularity for non-autonomous

evolution problems

In this section, we consider non-autonomous problems of the form

(t) + A(t)u(t) = f (t), t ≥ s,

u(s) = u

(5.1)

for some s ≥ 0. Compared with the problems studied before, the difference is now

that the operator A itself depends on the time t. The main consequence is that the

operators

and A do not commute anymore. We will present here results without

proofs (references for the proofs are however given).

5.1 Operator families depending smoothly on time

We will here assume that the operators (A(t))

t∈[0,T ]

deﬁned on a Banach space X are

uniformly (in t ∈ [0, T ]) sectorial for all t ∈ [0, T ] (see Deﬁnition A.10 below). This

implies in particular that −A(t) is the generator of an analytic semigroup in X for all

t ∈ [0, T ]. Moreover, we assume the so-called Acquistapace–Terreni condition: there

exist constants c > 0, α ∈ [0, 1[ and δ ∈]0, 1] such that

A(t)(λI + A(t))

−1

[A(t)

−1

− A(s)

−1

]

L (X)

≤

c|t −s|

1 + |λ|

1−α

(5.2)