Lecture Notes in Mathematics

Transcription

1 Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zorich 309 David H. Sattinger University of Minnesota, Minneapolis, MN/USA Topics in Stability and Bifurcation Theory Springer-Verlag Berlin Heidelberg New York 1973

2 AMS Subject Classifications (1970): 35-02, 35B35, 35G20, 35J60, 35K55, 35Q10, 46-xx, 76D05, 76E99 ISBN Springer-Verlag Berlin Heidelberg New York ISBN Springer-Verlag New York Heidelberg. Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, 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 the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer-Verlag Berlin - Heidelberg Library of Congress Catalog Card Number Printed in Germany. Offsetdmck: Julius Beltz, Hemsbach/Bergstr.

3 Preface In analyzing the dynamics of a physical system governed by nonlinear equations the following questions present themselves: Are there equilibrium states of the system? How many are there? Are they stable or unstable? What happens as external parameters are varied? As the parameters are varied, a given equilibrium may lose its stability (although it may continue to exist as a mathematical solution of the problem) and other equilibria or time periodic oscillations may branch off. Thus, bifurcation is a phenomenon closely related to the loss of stability in nonlinear physical systems. The subjects of bifurcation and stability have always attracted the interest of pure mathematicians, beginning at least with Poincar$ and Lyapounov. In the past decade an increasing amount of attention has focused on problems in partial differential equations. The purpose of these notes is to present some of the basic mathematical methods which have developed during this period. They are primarily mathematical in their approach, but it is hoped they will be of value to those applied mathematicians and engineers interested in learning the mathematical techniques of the subject. In a number of sections we have explained the basic mathematical tools needed for the development of the subject of bifurcation theory, for example elements of the theory of elliptic boundary value problems,

4 IV the Riesz-Schauder theory of compact operators~ the Leray-Schauder topological degree theory~ and the implicit function theorem in a B~uach space. Nevertheless, the reader will have to have a certain amotmt of background in mathematical analysis, particularly in the areas of partial differential equations and functional analysis, in order to benefit from these notes. These notes were the basis of a course given at the University of Minnesota during the academic year The author would like to express his thanks to Professors D. G. Aronson, Gene Fabes, Charles McCarthy, and Daniel Joseph for their lively interest in the course, and for their stimulating and cogent remarks. ~aanks are also due Miss L. Ruppert and Miss P. Williams for their fine job of typing. He would also like to acknowledge the support of the Air Force contract in the production of these notes. (AFOSR ) We hope these notes prove a convenient source of references and stimulate further interest in a diverse subject. D. H. Sattinger Minneapolis May i0, 1972

5 Contents I. Introduction... I Results from ordinary differential equations; examples of physical systems governed by partial differential equations. II. Nomlinear Ell2ptic Boundary. Value Problems of Second Order.. 14 Maximum principles. Function spaces. Existence theory of second order elliptic problems. Eigenvalue problems. Monotone iteration schemes. A simple bifurcation problem. An initial value problem. Stability. A singular perturbation problem. III. Functional Analys~s Banach Spaces. The Riesz-Schauder theory. Frechet derivatives. Implicit function theorem. Analytic operators. Decomposition of Vector Fields. IV. Bifurcation at a Simple Ei~envalue The Navier Stokes equations. Continuation of solutions. Bifurcation - Poinoar~-Lindstedt series. Stability of bifurcating solutions.

6 VI V. Bifurcation of Periodic Solutions Riesz-Schauder theory for a parabolic operator. Solution of the bifurcation problems. Formal stability of the bifurcating solutions; Floquet exponents. Examples from Chemical Reactor theory. VI. The Mathematical Problems of Hydrodynamic Stability Lyapounov's theorem for the Navier Stokes equations. VII. Topological Degree Theory and its Applications Finite dimensional degree theory. Leray-Schauder degree theory. Bifurcation by Leray-Schauder degree. Theorems of A mann and Rabinowitz. Vlll. The Real World Examples from hydrodynamics. B@nard and Taylor Problems.