A Problem Seminar (Donald J. Newman)

Abstract

just described it is clear that for every class of parabolic and hyperbolic equations one can seek appropriate functions and compatible conditions in the given data and the coefficients of the equation so that a concavity theorem results. Indeed, the current literature abounds with such results and the author has made a careful selection of important and interesting results of this type to include in the first seven chapters. The weighted energy method is also used for existence, nonexistence and asymptotic decay for solutions of differential equations and inequalities. Let L be a second order differential operator and let K; (x, t,/5), 1, 2, 3 be three functions depending on a real parameter/3. Then the weighted energy method consists in essence of obtaining an L2inequality of the form f f g _lOul <= f where D represents the first derivatives of u. If the kernels K; are selected strategically and the dependence of K; on the parameters/3 is carefully chosen, then the weighted energy inequality, as (**) is called, can be used to read off information about solutions of an equation of the form Lu f(x, t, u, Du).The author has gathered together applications of the above techniques to the Navier-Stokes equations, jet flow, bio-convection, plasma physics, magnetohydrodynamics, and decay estimates in relativistic theories. In each brief chapter a few details are given for one or two problems with references to the current literature for many other results. This book will serve as an excellent starting point for any worker interested in beginning a research program either in the general development of convexity and weighted energy methods or in the applications of these methods to a wide variety of physical theories. The bibliography of almost 200 papers is invaluable, not only to those interested in learning the subject but to the experienced workers who are interested in keeping current with the profusion of results in these areas.