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 275

holds for all t, s ∈ [0, T ] and λ ∈ Σ

= {z ∈ C \ {0}; |arg(z)| < π − θ} for some θ ∈

[0,

[. This condition implies in particular Hölder continuity in time of t 7→ A(t)

−1

Let us point out that the Acquistapace–Terreni condition allows however the domains

D(A(t)) to depend on t. This condition has been ﬁrst used by P. Acquistapace and

B. Terreni in [1] (and in a somewhat more abstract form by R. Labbas and B. Terreni

in [26] and [27]) to prove the existence of an evolution family (U (t, s))

t≥s, s,t∈[0,T ]

(for

all t ≥ s ≥ 0, U(t, s) is a bounded operator in X) so that u(t) = U(t, s)u

, t ≥ s, is

the solution of (5.1) with f = 0. The general form of a solution of (5.1) is then given

by the formula

u(t) = U(t, s)u

U(t, r)f (r) dr, t ≥ s .

Deﬁnition 5.1. The family of operators (A(t))

t∈[0,T ]

is said to have the maximal L

regularity property if, for all f ∈ L

(0, T ; X), there exists a unique solution u of (5.1)

(with s = 0 and u

= 0) satisfying u

∈ L

(0, T ; X) and

u(t) ∈ D(A(t)) a.e. in [0, T ] 3 t and t 7→ A(t)u(t) ∈ L

(0, T ; X).

As in the autonomous case (see Proposition 2.4), the property of maximal L

regularity is independent of p ∈]1, ∞[ under the condition (5.2).

Theorem 5.2. Let X be a Banach space. Let A = (A(t))

t∈[0,T ]

be a family of uni-

formly sectorial operators on X satisfying the Acquistapace–Terreni condition (5.2).

Assume that A enjoys the maximal L

-regularity property for one p ∈]1, ∞[. Then A

has the maximal L

-regularity property for all q ∈]1, ∞[.

Reference for the proof. The proof of this theorem is due to M. Hieber and S. Monni-

aux and can be found in [20, Theorem 3.1]. The idea of the proof is to show that the

condition (5.2) implies a Hörmander-type condition for the operator S deﬁned by

Sf(t) =

A(t)e

−(t−s)A(t)

f(s) ds, t ∈ [0, T ],

which is the singular part of the operator f 7→ A(·)u(·) for u being the solution of (5.1)

with s = 0 and u

= 0. Therefore, we can apply Theorem 2.5. Since S is bounded in

(0, T ; X) for one p ∈]1, ∞[, it is also bounded in L

(0, T ; X) for all q ∈]1, ∞[.

In the case of Hilbert spaces, we have the following result.

Theorem 5.3. Let X = H be a Hilbert space. Let A = (A(t))

t∈[0,T ]

be a family

of uniformly sectorial operators on H satisfying the Acquistapace–Terreni condition

(5.2). Then A enjoys the maximal L

-regularity property for all p ∈]1, ∞[.

Reference for the proof. The proof is due to M. Hieber and S. Monniaux and can be

found in [20, Theorem 3.2]. It appears as a corollary of Theorem 5.2 and Theorem 5.4

below with the symbol a deﬁned by

a(t, τ) =











A(0)(iτI + A(0))

−1

, t < 0,

A(t)(iτI + A(t))

−1

, t ∈ [0, T ],

A(T )(iτI + A(T ))

−1

, t > T.

276 Sylvie Monniaux

Indeed, it sufﬁces to show that a satisﬁes the conditions of Theorem 5.4 to get the

maximal L

-regularity property of A and it remains to apply Theorem 5.2 to obtain

the maximal L

-regularity property for all p ∈]1, ∞[.

Theorem 5.4. Let H be a Hilbert space and a ∈ L

∞

(R × R; L (H)) such that ξ 7→

a(x, ξ) has an analytic extension (with values in L (H)) in S

= {±z ∈ C; |arg(z)| <

θ} for one θ ∈]0,

[ and

sup

z∈S

sup

x∈R

ka(x, z)k

L (H)

< ∞.

Let u ∈ S (R; H) and deﬁne

Op(a)u(x) =

√

2π

ixξ

a(x, ξ)F (u)(ξ) dξ, x ∈ R.

Then Op(a) extends to a bounded operator on L

(R; H).

Reference for the proof. The proof of this theorem is due to M. Hieber and S. Monni-

aux (see [20, Theorem 2.1]). This can be viewed as a parameter dependent version of

the Fourier-multiplier theorem in Hilbert spaces.

A version of Theorem 5.4 was proved by P. Portal and Ž. Štrkalj in UMD-spaces

(see [38]). This allows to prove the following result, similar to Theorem 5.3 in the

case of UMD-spaces (see also [41] for the autonomous case). The idea is to replace

boundedness of the resolvent in the case of a Hilbert space with R-boundedness of the

resolvent in the case of a Banach space with the UMD-property.

Deﬁnition 5.5. Let X be a Banach space. Let A = (A(t))

t∈[0,T ]

be a family of uni-

formly sectorial operators on X. Then A is said to be a family of uniformly R-sectorial

operators on X if it satisﬁes

{(1 + |λ|)(λI + A)

−1

; λ ∈ Σ

, t ∈ [0, T ]}

< ∞,

where the R-bound R(τ) of a set of bounded operators τ has been deﬁned in Section 2

(Deﬁnition 2.12).

Theorem 5.6 (Portal-Štrkalj, 2006). Let X be a UMD-space. Let A = (A(t))

t∈[0,T ]

a family of uniformly R -sectorial operators on X satisfying the Acquistapace–Terreni

condition (5.2). Then A enjoys the maximal L

-regularity property for all p ∈]1, ∞[.

Reference for the proof. The proof of this result can be found in [38, Corollary 14 in

Section 5].

The ﬁrst theorem about maximal regularity in the non-autonomous setting was

proved by S. Monniaux and J. Prüss in 1997 (see [34]) and is a generalization of

the theorem of Dore–Venni (see Theorem A.14 below) when the operator A depends

on t and satisﬁes a condition of Acquistapace–Terreni-type (see (5.2)).

Maximal regularity and applications to PDEs 277

Theorem 5.7. Let X be a UMD-Banach space. Let A = (A(t))

t∈[0,T ]

be a family

of uniformly sectorial operators with bounded imaginary powers on X satisfying the

Acquistapace–Terreni condition (5.2) such that

sup

t∈[0,T ]

sup

s∈R

|s|

ln kA(t)

L (X)

∈

Then A enjoys the maximal L

-regularity property for all p ∈]1, ∞[.

Reference for the proof. This result is due to S. Monniaux and J. Prüss and its proof,

in a slightly more general setting, can be found in [34, Theorem 1]. Remark that in

particular, the bound on {A(t)

; s ∈ R} implies by Theorem A.14 that every operator

A(t) has the property of maximal L

-regularity.

The result presented now is a generalization of Theorem 3.3 in the non-autonomous

setting.

Theorem 5.8. Let Ω ⊂ R

and A = (A(t))

t∈[0,T ]

be a family of unbounded operators

deﬁned on L

(Ω) such that −A(t) generates an analytic semigroup in L

(Ω). We

assume moreover that A satisﬁes the Acquistapace–Terreni condition (5.2) in L

(Ω)

and that for all t ∈ [0, T ] the semigroup (e

−sA(t)

)

s≥0

has a kernel p

(s, x, y) with

uniform Gaussian upper bounds, more precisely,

−sA(t)

f(x) =

Ω

(s, x, y)f(y) dy, x ∈ Ω, t ∈ [0, T ], s ≥ 0,

and there exist constants c, b > 0 (independent of t) such that

(s, x, y)| ≤ cg(bs, x, y), x, y ∈ Ω, t ∈ [0, T ], s ≥ 0,

where g was deﬁned in Theorem 3.3. Then A enjoys the maximal L

-regularity prop-

erty in L

(Ω) for all p, q ∈]1, ∞[.

Reference for the proof. This result is due to M. Hieber and S. Monniaux and its proof,

though in a slightly more general setting, can be found in [19, Theorem 1]. Remark

that in particular, the Gaussian bound for p

implies by Theorem 3.3 that every operator

A(t) has the property of maximal L

-regularity.

5.2 Operator families with domain constant in time and

quasilinear problems

The case where the domains D(A(t)) of the operators A(t), t ∈ [0, T ] do not depend

on t, i.e., D(A(t)) = D ⊂ X, was investigated by J. Prüss and R. Schnaubelt [39],

H. Amann [2], in view of applications to quasilinear evolution equations in [3], and by

W. Arendt, R. Chill, S. Fornaro and C. Poupaud [5].

To state the main result in its full generality, we need to introduce the notion of

relative continuity.

278 Sylvie Monniaux

Deﬁnition 5.9. A function A : [0, T ] → L (D, X) is called relatively continuous if for

each t ∈ [0, T ] and all ε > 0 there exist δ > 0 and η ≥ 0 such that for all x ∈ D,

kA(t)x −A(s)xk

≤ εkxk

+ ηkxk

holds for all s ∈ [0, T ] with |t −s| ≤ δ.

Theorem 5.10. Let A : [0, T ] → L (D, X) be strongly measurable and relatively con-

tinuous. Assume that A(t) has the maximal L

-regularity property for all t ∈ [0, T ].

Then for each x ∈ (X, D)

and f ∈ L

(0, T ; X) there exists a unique solution u of

(5.1) satisfying u ∈ L

(0, T ; D) ∩W

1,p

(0, T ; X).

References for the proof. This theorem was ﬁrst proved by J. Prüss and R. Schnaubelt

in the case where A : [0, T ] → L (D, X) is continuous (see [39, Theorem 2.5] and

also [2, Theorem 7.1]). They proved in particular that the hypotheses of the theorem

imply the existence of an evolution family (U(t, s))

t≥s, s,t∈[0,T ]

. The condition x ∈

(X, D)

is to compare with Remark 2.3. The theorem, in its full generality as stated

above, is due to W. Arendt, R. Chill, S. Fornaro and C. Poupaud (see [5, Theorem 2.7]).

They also proved the existence of an evolution family (U (t, s))

t≥s, s,t∈[0,T ]

and the

solution u of (5.1) is given by

u(t) = U(t, s)u

U(t, r)f (r) dr, t ≥ s.

Non-autonomous maximal regularity results seem to be a good starting point to

study quasilinear evolution equations. This has been thoroughly studied by H. Amann

(see, e.g., [3]). The problem is the following: ﬁnd a solution u of

+ A(u)u = F (u) on [0, T ], u(0) = u

, (5.3)

with “reasonable” conditions on u 7→ A(u) and u 7→ F (u). The idea to treat this

problem is to apply a ﬁxed point theorem on the map

v 7→ u, where u

(t) + A(v(t))u(t) = F (v(t)), t ∈ [0, T ], u(0) = u

. (5.4)

The problem is of course to ﬁnd a suitable Banach space in which one can apply the

ﬁxed point theorem, i.e., for which the solution of (5.4) has the best possible regularity

properties, such as maximal regularity. The situation studied in [3] is the following.

Let

= L

(0, T ; D) ∩W

1,p

(0, T ; X)

be the space associated to maximal L

-regularity for (5.4). The operators A(u) and the

function F satisfy, for all u ∈ E

A(u) ∈ L

∞

(0, T ; L (D, X)) and F (u) ∈ L

(0, T ; X).

It is also assumed that the restriction of (A(u), F (u)) on a subinterval J depends only

on the restriction of u on the interval J (i.e., A and F are Volterra operators).

Maximal regularity and applications to PDEs 279

Theorem 5.11. Under the above assumptions, the problem (5.3) has a unique maximal

solution.

Reference for the proof. This result is due to H. Amann [3, Section 2].

A Appendix

We collect here some of the results used in the previous sections, mostly without proofs

but with references where those can be found.

A.1 Picard’s ﬁxed point theorem for bilinear operators

Let F be a Banach space and let a ∈ F . We want to solve the equation

u = a + B(u, u), u ∈ F , (A.1)

where B is a symmetric bilinear continuous operator on F × F . Let kBk be the

smallest constant c > 0 such that

kB(u, v)k

≤ ckuk

kvk

, u, v ∈ F .

Theorem A.1 (Picard’s ﬁxed point theorem). For all a ∈ F with kak

≤

4kBk

, there

exists a solution u ∈ F of (A.1). More precisely, there exists for all k = 1, 2, . . .

an operator T

, the restriction on the diagonal of a continuous k-linear operator from

F × . . . × F to F , satisfying

(i) there exists a constant c > 0 (even not depending on F ) such that for all a ∈ F

and all k = 1, 2, . . .

(a)k

≤

kBk

−

(4kBkkak

)

;

(ii) u has the form

u =

∞

k=1

(a).

Moreover, we have kuk

≤

2kBk

, and u is the unique solution of (A.1) in the closed

ball B

(0,

2kBk

Proof. We set T

(a) = a and deﬁne by induction

(a) =

k−1

`=1

B(T

(a), T

k−`

(a)). (A.2)

280 Sylvie Monniaux

The fact that T

, k = 1, 2, . . . , is the restriction of a k-linear operator is clear by

induction. By induction, we can also prove that for all k = 1, 2, . . . , there exists a

constant α

> 0 such that kT

(a)k

≤ α

kak

. By the deﬁnition of T

, we have

(a)k

≤ kBk

k−1

`=1

(a)k

k−`

(a)k

We can then chose α

deﬁned by induction by

= 1, α

= kBk

k−1

`=1

k−`

, k = 2, 3, . . . .

The numbers c

= kBk

−k+1

, k = 1, 2, . . . , satisfy

= 1, c

k−1

`=1

k−`

, k = 2, 3, . . . .

These numbers are Catalan’s numbers with the explicit representation c

(2k−2)!

k!(k−1)!

. It

is known that there exists a constant c > 0 such that c

≤ c 4

−

for all k = 1, 2, . . . ,

which gives the bound announced in (i). It is also known that

∞

k=1

1 −

√

1 −4z

, |z| <

. (A.3)

Let now a ∈ F with kak

≤

4kBk

. Then the series

∞

k=1

(a) is uniformly conver-

gent in F with limit u. We will prove now that u is the solution of (A.1). Indeed, we

have

B(u, u) = B

∞

`−1

(a),

∞

m=1

(a)

∞

`,m=1

B(T

(a), T

(a))

∞

k=2

`+m=k

B(T

(a), T

(a)) =

∞

k=2

(a) = u − T

(a) = u − a.

The ﬁrst equality comes from the deﬁnition of u. The second equality is due to the

bilinearity of B. The third equality is obtained by rearranging the double sum which

is allowed because of the uniform convergence of the series. The fourth equality uses

only the deﬁnition (A.2) of T

. The last two equalities are obvious. In addition, we

have

kuk

≤

∞

k=1

(a)k

≤

∞

k=1

kBk

k−1

kak

≤

2kBk

(1 −

1 −4kBkkak

) ≤

2kBk

Maximal regularity and applications to PDEs 281

The ﬁrst inequality follows from the deﬁnition of u. The second inequality comes from

the estimates kT

(a)k

≤ c

kBk

k−1

kak

for all k = 1, 2, . . . . The third inequality

is in fact an equality coming from the formula (A.3). The last inequality is obvious.

So far, we have proved the existence of a solution u ∈ F of (A.1) for a ∈ F with

kak

≤

4kBk

. We also proved that kuk

≤

2kBk

. It remains to prove that such a

solution is unique in the closed ball B

(0,

2kBk

). Assume that there exists another

solution v of (A.1) in B

(0,

2kBk

). We can distinguish two cases:

(i) If kuk <

2kBk

or kvk <

2kBk

, then, by taking the difference of u = a + B(u, u)

and v = a + B(v, v), we obtain u −v = B(u − v, u + v) and therefore

ku −vk

≤ kBkku −vk

ku + vk

≤ kBk(kuk

+ kvk

)ku −vk

< ku − vk

which implies directly that u = v.

(ii) If kuk

2kBk

and kvk

2kBk

then for all n ≥ 1, we deﬁne

= v − T

(a) −. . . − T

(a).

By induction, we can prove that

≤

∞

k=n+1

kBk

−k

−−−−→

n→∞

and therefore v =

∞

k=1

(a) = u.

This completes the proof of Theorem A.1.

A.2 Interpolation of operators

Theorem A.2 (Riesz–Thorin theorem). Let (X, X, µ) and (Y, Y, ν) be two ﬁxed mea-

sure spaces. Let 1 ≤ p

, p

, q

≤ ∞ and θ ∈]0, 1[. Let

T : L

(X) + L

(X) → L

(Y ) + L

(Y )

be a linear operator such that there exist two constants c

, c

> 0 with

kT fk

(Y )

≤ c

kfk

(X)

for all f ∈ L

(X), i = 0, 1.

Then we have for all f ∈ L

(X)

kT fk

(Y )

≤ c

kfk

(X)

where

1−θ

and c

= c

1−θ

282 Sylvie Monniaux

References for the proof. A proof of this theorem can be found in [8, Theorem 1.1.1].

Another nice reference is T. Tao’s lecture on this subject (see [40, Theorem 3]).

Deﬁnition A.3. An operator T mapping measurable functions over a measure space

(X, X, µ) into measurable functions over a measure space (Y, Y, ν) is said to be sub-

linear if it maps simple functions f : X → C, whose support is of ﬁnite measure, to

nonnegative-valued functions (modulo almost everywhere equivalence) such that the

homogeneity

T (cf) = |c|T f for all c ∈ C

and the pointwise sublinearity

|T (f + g)| ≤ |T f| + |T g|

are satisﬁed for all simple functions f, g whose support is of ﬁnite measure.

Remark A.4. If S is a linear operator, then T = |S| is sublinear.

Deﬁnition A.5. A linear or sublinear operator mapping measurable functions over a

measure space (X, X, µ) into measurable functions over a measure space (Y, Y, ν)

is said to be of strong type (p, q) if there exists a constant c > 0 such that for all

f ∈ L

(X)

kT fk

(Y )

≤ ckfk

(X)

It is said to be of weak type (p, q) if there exists a constant c > 0 such that for all

f ∈ L

(X)

sup

t≥0

tµ

{x ∈ X; |T f(x)| ≥ t}

≤ ckfk

(X)

Theorem A.6 (Marcinkiewicz interpolation theorem). Let 1 ≤ p

, p

, q

≤ ∞ and

θ ∈]0, 1[ such that q

6= q

and p

≤ q

for i = 0, 1. Let T be a sublinear operator of

weak type (p

, q

) and of weak type (p

, q

). Then T is of strong type (p

, q

) where

1−θ

and

1−θ

References for the proof. A proof for this theorem can be found in [8, Theorem 1.3.1].

Another nice reference is T. Tao’s lecture on this subject [40, Theorem 4].

Theorem A.7 (Mikhlin multiplier theorem). Let X and Y be Hilbert spaces. If the

differentiable function M : R \ {0} → L (X, Y ) satisﬁes

kM(t)k

L (X,Y )

≤ C and ktM

(t)k

L (X,Y )

≤ C for all t ∈ R \ {0}

for some constant C > 0, then M is a Fourier multiplier (see Deﬁnition 2.11) in

(R; X) for all p ∈]1, ∞[.

References for the proof. See [8, Section 6.1].

Maximal regularity and applications to PDEs 283

A.3 Calderón–Zygmund theory

Theorem A.8 (Calderón–Zygmund decomposition). Let f ∈ L

) and ﬁx λ > 0.

Then f = g +

, where

(i) |g| ≤ 2

λ almost everywhere;

(ii) kgk

≤ 3kfk

;

(iii) there exists a family of disjoint cubes (Q

)

k∈N

of R

such that

supp b

⊂ Q

dx = 0 and

| ≤

kfk

Proof. Let f ∈ L

) and ﬁx λ > 0. We may assume kfk

= 1. We decompose R

into cubes of measure

: R

0,m

. Then we have for all m

0,m

|f|dx ≤ λkfk

= λ.

We then decompose each cube

0,m

into cubes of measure

. We denote by (Q

1,m

)

all cubes for which

1,m

|f|dx > λ.

The other cubes are denoted by

1,m

. We repeat this operation with these cubes

1,m

and obtain cubes (Q

2,m

)

of measure

for which

2,m

|f|dx > λ,

the remaining cubes being denoted by

2,m

. After a countable number of steps, we

obtain a family of cubes (Q

j,m

)

j,m∈N

renamed as (Q

)

k∈N

. We now deﬁne b

= (f − m

(f))χ

with the notation

(f) =

|Q|

f dx.

We denote by g the quantity

g = f −

It remains to show that g and the b

’s satisfy the conditions (i), (ii) and (iii) of the

theorem. First, if x ∈ R

, then for all j ∈ N, there exists m ∈ N such that

x ∈

j,m

. By construction, we have

λ ≥

j,m

|f|dy −−−→

j→∞

|f(x)|

284 Sylvie Monniaux

if x is a Lebesgue point of f . If x ∈ Q

for one k ∈ N, then

|g(x)| = |m

(f)| ≤

λ = 2

λ.

Altogether, this gives (i) since the set of points that are not Lebesgue points of f is of

measure zero. Again by construction, we have

supp b

⊂ Q

and

dx = 0.

Moreover, we have

|f|dx > λ. Therefore, we ﬁnd |Q

| <

|f|dx. Since

the cubes Q

are disjoint, this gives

| ≤

|f|dx ≤

|f|dx =

kfk

This proves (iii). Finally, we have kgk

≤ kfk

and kb

≤ 2

|f|dx for all k ∈ N,

and this gives (ii).

Remark A.9. The Calderón–Zygmund decomposition is also available in a measurable

space (E, µ, d) of homogeneous type where µ is a σ-ﬁnite measure and d is a quasi-

metric, i.e., there exists a constant c > 0 such that for any ball B = {y; d(x, y) < r},

if we denote by 2B the ball with the same center and twice the radius of B, it holds

µ(2B) ≤ cµ(B).

A.4 Bounded imaginary powers

Deﬁnition A.10. A (linear) operator A on a Banach space X is sectorial if it is closed,

densely deﬁned, has empty kernel, dense range R(A) and satisﬁes

sup

t>0

kt(tI + A)

−1

L (X)

< ∞.

Let x ∈ D(A) ∩ R(A). Then one can deﬁne for z ∈ C with |Re(z)| < 1

x =

sin πz

−

1 + z

−1

x +

z+1

(tI + A)

−1

x dt

∞

z−1

(tI + A)

−1

Ax dt

We are now in the position to give the deﬁnition of operators with bounded imagi-

nary powers.

Deﬁnition A.11. A sectorial operator on a Banach space X has bounded imaginary

powers if the closure of the operator (A

, D(A) ∩ R(A)) deﬁnes a bounded operator

on X for all s ∈ R and if sup

|s|≤1

L (X)

< ∞.