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 285

Remark A.12. If A admits bounded imaginary powers, then (A

)

s∈R

forms a strongly

continuous group on X.

Remark A.13. Conversely, it has been proved in [32] that a given strongly continuous

group of type strictly less than π on a UMD-Banach space can be represented as the

imaginary powers of a sectorial operator, called its analytic generator.

In the class of UMD-spaces, a positive result for operators having bounded imagi-

nary powers has been proved by G. Dore and A. Venni.

Theorem A.14 (Dore–Venni, 1987). Let X be a Banach space in the UMD-class. Let

A be an operator with bounded imaginary powers (see Deﬁnition A.11) for which the

type of the group (A

)

s∈R

is strictly less than

. Then A has the maximal L

-regula-

rity property.

Idea of the proof. The idea is to show that A+ B with domain D (A) ∩D(B) is invert-

ible, where

D(A) = L

(0, ∞; D(A)), (Au)(t) = Au(t), t > 0,

and

D(B) = H

(0, ∞; X), Bu = u

The operator A has bounded imaginary powers with angle strictly less than

and the

operator B has bounded imaginary powers with angle

. We deﬁne then, for c ∈]0, 1[,

S =

c+i∞

c−i∞

−z

z−1

sin πz

dz.

The purpose is to show that BS is bounded and that S = (A + B)

−1

, and this is done

by letting c → 0

and taking into account that the Hilbert transform is bounded in

(R; X).

Another (shorter) proof uses the result presented in Remark A.13 by showing that the

group (A

−is

)

s∈R

has a sectorial analytic generator.

Acknowledgments. The author would like to thank the Technische Universität Berlin

where the content of this survey has been presented in a series of four lectures during

May 2009. Special thanks go to Petra Wittbold and Etienne Emmrich for their kind

invitation and to the students and colleagues who attended the course.

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Author information

Sylvie Monniaux, LATP UMR 6632 – Case Cour A – Faculté des Sciences de Saint-Jérôme – Aix-

Marseille Université – Avenue Escadrille Normandie Niémen – 13397 Marseille, France.

E-mail: sylvie.monniaux@univ-cezanne.fr

Index

Acquistapace–Terreni condition, 274

admissibility condition, 32

jump, 32

analytic semigroup, 249

Arlequin method, 228

boundary entropy-entropy ﬂux pair, 126

bounded imaginary powers, 284

Burgers equation, 35

Calderón–Zygmund decomposition, 251, 283

canonical ensemble, 232

Cauchy–Born rule, 200, 225

characteristic, 5

characteristic ﬂow

approximate, 79

exact, 73

conservation of

charge, 154

energy, 154

mass, 3, 154

momentum, 34, 154

consistency, 195

constitutive law, 197

contraction semigroup, 259

coupled energy, 202, 203, 206

crack propagation, 194

deformation, 197

discontinuity

strong, 17

weak, 17

Dore–Venni theorem, 276

elastic material, 197

entropy process solution, 132

entropy solution, 126

generalized, 30

error estimate

for adaptive schemes, 111

for uniform schemes, 83

evolution family, 275

ﬁnite element tree adaptation

algorithm of, 106

deﬁnition of, 89

ﬂux function, 9

convex, 31

with inﬂexion point, 55

Fourier multiplier, 257

free energy, 233

Gaussian estimates, 262

generalized, 264

Grillakis–Shatah–Strauss theory, 170

ground state, 163

Hörmander condition, 251

Hilbert transform, 255

Holmgren-type duality method, 124

Hopf equation, 2

instability by blow-up, 178

interface condition, 206, 207

irreversibility and entropy increase, 40

Kruzhkov entropies, 49

Lax condition, 33

Leray projection, 272

local minimizer, 196, 224

Marcinkiewicz interpolation theorem, 253, 282

maximal L

-regularity, 248

method of doubling variables, 127

Mikhlin multiplier theorem, 257, 282

mountain pass theorem, 160

nanoindentation, 194

Navier equation, 234

Navier–Lamé operator, 265

Navier–Stokes equations, 271

Nehari manifold, 158

non-autonomous evolution problem, 274

Ole

ınik inequality, 32

orbital stability, 166

p-system, 62

290 Index

partition of domain (macro/microscopic), 202,

206

“physically correct” solution, 30

Picard’s ﬁxed point theorem, 279

Pohozhaev identity, 158

prediction of adaptive trees, 108

QuasiContinuum method, 224

R-boundedness, 257

Rankine–Hugoniot condition, 20, 28

rarefaction wave, 52

regular entropy pair, 125

Riemann problem, 50

Riesz–Thorin theorem, 281

Schauder–Tikhonov ﬁxed point theorem, 118

sectorial operator, 275, 284

self-similar solution, 50

semi-Lagrangian method, 77

semilinear heat equation, 268

shock wave, 21

standing wave, 157

thermodynamic limit, 201, 233

transference principle, 259

UMD-space, 254

vanishing viscosity, 34, 132

Vlasov–Poisson system, 71

Volterra operator, 278

wavelet tree adaptation

algorithm of, 106

deﬁnition of, 105