Weighted Energy Methods in Fluid Dynamics and Elasticity
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1134 Giovanni R Galdi Salvatore Rionero
Weighted Energy Methods in Fluid Dynamics and Elasticity
Springer-Verlag Berlin Heidelberg New York Tokyo
Authors
Giovanni P. Galdi Salvatore Rionero Dipartimento di Matematica "R. Caccioppoli" Mezzocannone 8 80134 Naples, Italy
Mathematics Subject Classification: 73C 10, 73C 15, 76E25, 76E30 ISBN 3-540-15645-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15645-3 Springer-Verlag New York Heidelberg BerlinTokyo
Library of Congress Cataloging-in-Publication Data. Galdi, Giovanni P. (Giovanni Paolo), 1967Weighted energy methods in fluid dynamics and elasticity. (Lecture notes in mathematics; 1134) Bibliography: p. Includes index. 1. Fluid dynamics. 2. Elasticity. 3. Differential equations- Numerical solutions. I. Rionero, Salvatore. II. Title. III. Series: Lecture notes in mathematics (Springer-Verlag); 1134. QA3.L28 no. 1134510 s 85-12662 [QA911] [532'.05] ISBN 0-387-15645-3 (U. S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translating, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE As is known, by energy methods (EM) one denotes a suitable mathematical device for deriving estimates of solutions to differential equations.
The name of the
method is due to the fact that it is usually founded upon "conservation laws" which must be obeyed by solutions.
A typical example of EM, and maybe one of the earliest,
is given by the approach introduced at the end of the nineteenth century by the Russian mathematician A.M. Liapounov for studying the stability of solutions to ordinary differential equations.
Actually, this approach is based on the existence
of a suitable functional which must be positive and non-increasing in time (the socalled "Liapounov functional").
Another no
less important example of EM is furnish-
ed by the method proposed by J. Serrin in 1959 and successively generalized and deepened by D.D. Joseph and his co-workers, for studying nonlinear stability of viscous incompressible flows in bounded domains.
Roughly speaking, the method con-
sists in forming the "kinetic energy" of perturbations to a given basic flow and in studying its behavior in time (see Chapter I).
Of course, there are several other
examples of applicability of EM, such as uniqueness, existence, etc., and for some of them the reader is referred to the papers cited throughout these Notes. Aside from the above EM, there are the so-called weighted energy methods (WEM). Unlike the former, the latter explicitly involve the use of suitable auxiliary functions (weight functions) whose task may vary from ca
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