Table Of ContentAspects of mathematics
Wolf von Wahl
The Equations of
Navier-stokes and
Abstract Parabolic
Equations
Wolf von Wahl
The Equations of Navier-Stokes
and Abstract Parabolic Equations
Aspects cl Mathernatics
Aspekte der Mathematik
Herausgeber Klas Diederich
Vol. E1: G. Hector/U. Hirsch, Introduction to the
Geometry of Foliations, Part A
Vol. E2: M. Knebusch/M. Koister, Wittrings
Vol. E3: G. Hector/U. Hirsch, Introduction to the
Geometry of Foliations, Part B
Vol. E4: M. Laska, Elliptic Curves over Number Fields
with Prescribed Reduction Type
Vol. E5: P. Stiller, Automorphic Forms and the Picard
Number of an Elliptic Surface
Vol. E6: G. Faltings/G. Wüstholz et al., Rational Points
(A Publication of the Max·Planck·1 nstitut für Mathematik, Bonn)
Vol. E7: W. StoII, Value Distribution Theory for
Meromorphic Maps
Vol. ES: W. von Wahl, The Equations of Navier-Stokes
and Abstract Parabolic Equations
Band 01: H. Kraft. Geometrische Methoden in der
Invariantentheorie
Die in dieser Reihe veröffentlichten Texte wenden sich an graduierte Studenten
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Die Reihe umfaßt zwei Unterreihen, eine deutsch· und eine englischsprachige.
Wolf von Wahl
The Equations of Navier
Stokes and Abstract
Parabolic Equations
Springer Fachmedien Wiesbaden GmbH
Prof. Dr. Wolf von Wahl is Professor of Applied Mathematics at the
University of Bayreuth, Fed. Rep. of Germany.
AMS Subject Classification: 35 Q 10, 35 K 22
1985
All rights reserved
© Springer Fachmedien Wiesbaden 1985
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig in 1985.
No part of this pUblication may be reproduced, stored in a retrieval system or transmitted
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise,
without prior permission of the copyright holder.
Produced by Lengericher Handelsdruckerei , Lengerich
ISBN 978-3-528-08915-3 ISBN 978-3-663-13911-9 (eBook)
DOI 10.1007/978-3-663-13911-9
Contents
O. Introduction, Auxiliary Propositions and Notations VII
§ 1. Introduction VII
§ 2. Auxiliary Propositions and Notations XII
I. Linear Equations of Parabolic Type
§ 1. Analytic Semigroups
§ 2. The Evolution Operator U(t,s) 4
§ 3. Fractional Powers 9
9 4. Cornrnents to Chapter I 12
11. Local Solutions of First Order Semilinear
Evolution Equations 18
§ 1. Solutions of Equations with Nonlinearities
Relatively Bounded to A 18
§ 2. A Nonlinear Interpolation Theorem 29
§ 3. Solutions of Equations with Nonlinearities
Relatively Bounded to A1-p, their Higher
Regularity and the Question of Admissible
Initial Data 37
§ 4. Cornrnents to Chapter 11 64
111. Local Solvability of the Equations of Navier-Stokes 67
§ 1. Solonnikov's Results for the Instationary
Stokes Equation 67
§ 2. Fractional Powers of the Stokes-Operator 80
§ 3. Local Strong Solvability of the Navier-Stokes
Equations 99
v
§ 4. Global Existence for Small Data. Extension
of the Previous Results to Arbitrary Dimensions 131
§ 5. Comments to Chapter III 136
IV. Global Existence and Global Regularity for the
Navier-Stokes Equations 139
§ 1. Weak Solutions 139
§ 2. Some Additional Regularity Properties for
Weak Solutions in General 149
§ 3. On the Validity of the Energy Inequality and
on the Regularity of the Expression u'+VTI 154
§ 4. On the Uniqueness of Weak Solutions. The
Connection between Weak Solutions and Local
Strong Solutions 166
§ 5. Regularity of Weak Solutions. Leray's
Structure Theorem 190
§ 6. Comments to Chapter IV 224
V. Global Solutions of Abstract Nonlinear Parabolic
Equations and Applications 226
§ 1. Abstract Nonlinear Parabolic Equations 226
§ 2. Applications to Parabolic Systems and to the
Equations of Navier-Stokes 235
§ 3. Comments to Chapter V 253
VI. References 255
VI
0. Introduction, Auxiliary Propositions,
and Notations
§ 1. Introduction
This book is mainly devoted to the study of the initial
boundary value problem for the system of Navier-Stokes
dU
{-dt - vAU + u·'Vu + 'V TI f,
(0. 1 . 1 )
'V·u 0,
(0.1.2) u(t,x) =0, t >0, x E an,
(0.1.3) u(O,x) =(j)(x)
over a cylindrical domain (O,T) xn c IR n+1. This system describes
the velocity u and pressure of a viscous incompressible fluid
TI
under the influence of an external force f. The viscosity v is
assumed to be constant. Of physical importance are only the
cases n = 3 and n = 2 (if the data depend on two variables only) .
Because of its mathematical interest however, we intend to
develop a theory for arbitrary n. In most cases the theory
available for n = 3 can be carried over to any n ;: 3 (after
suitable modifications). n is the space domain filled out by
the fluid; it is assumed to be bounded although many of our
results remain valid for the cases that n is the whole of IRn
or an exterior domain.
The access to (0.1.1) is a functional analytic one. There
fore we treat in great detail the local strong solvability of
abstract parabolic equations
VII
u' +Au +M(u) 0,
(0.1.4) {
u(O)
in a Banach space B. Here-A generates an analytic semigroup in
Band M is a nonlinear mapping fulfilling suitable Lipschitz
conditions. In particular we deal with the question which ini
tial data are admissible in order to guarantee the unique
existence of a sufficiently "nice" solution in some maximal
interval (O,T(~)). Also the question how to characterise the
case T(~) <+= is studied in detail. This theory is then applied
to (0.1.1) by making use of the well known fact that (0.1.1)
can be transformed into an equation like (0.1.4) if we apply
formally the projection P of some (LP(Q))n onto its divergence
p
free part H (Q). Then we get
p
u' - vP Llu + P (u·'Vu) P f,
P P P
u(O) = <.p,
where ~ is assumed to be divergence free in some generalized
sense. The negative Stokes-operator -A=vP Ll turns out to be
p
the generator of an analytic semigroup in the Banach space H (Q)
P
and M(u) =P (u·'Vu) fulfills Lipschitz conditions being covered
p
by our abstract theorYi of course we have assumed that, because
of 'V·u(t,x) =0 in (0.1.1), the function u is invariant under P .
P
As it is well known, besides this abstract setting one can
construct a possibly non-unique and non-regular weak solution
to (0.1.1) for all times and any space dimension ni this re
sult goes back to E. Hopf [Ho]. Following the ideas of Leray
[Ler] and Serrin [Ser2] one main concern of this book is to
study the connections between local (in t) strong solutions and
weak solutions. Thus we get in particular results concerning
the possible singularities (in t) of weak solutions.
VIII
The second aim of the book is to study criteria for the
uniqueness and global (in t) regularity of weak solutions. For
that purpose we have not only to use our abstract theory but
also to rely on a fundamental idea of Serrin [Ser1, Ser21:
Serrin has studied the consequences of the assumption that a
weak solution. u of (0.1.1) is in some space LS ((O,T), (Lr (r2) )n)
for suitable s,r. For that purpose he has introduced a "criti
cal quantity" q(s,r) =~+Q. We deal in detail with q(s,r) and,
s r
in particular, we try to cover all marginal values of q(s,r)
which were not considered by Serrin.
In order to get results as best as possible for the global
regularity it is not surprising that we need so called maximal
regularity results for the linear part of (0.1.1). The esti
mates being necessary for that aim are not provided by the
theory of semigroups but by the potential theoretical estimates
of Solonnikov [Sol 1,21. A consequence of these estimates is
e. g. the fact that -A = vP /::,. generates an analytic semigroup
p
inH(I1).
p
In general one can say that the best results are furnished
by an interaction between the various methods to solve (0.1.1).
In chapter V we use abstract methods and estimates for the
concrete equations to prove both, the global existence of solu
tions to certain parabolic systems and the regularity of weak
solutions of the Navier-Stokes equations being in CO([0,T1,
(Ln (I1))n). The abstract method employed here for equations
like (0.1.4) gives a very precise description of what happens
when the length of the maximal interval of existence is finite;
this enables us to study the question of global existence for
semilinear parabolic systems
ul ' + L Da(Aaß(X)Dßul) + fl(t,x,u) 0, 1 ;;;; 1 ;;;; L,
I al ;;;;m,
I I
ß ;;;;rn
IX
