Handbook of Differential Equations Evolutionary Equations Vol I

Abstract

The aim of this Handbook is to acquaint the reader with the current status of the theory of evolutionary partial differential equations, and with some of its applications. This is not an easy task: Unlike other mathematical theories which exhibit a tree-like structure, with clearly distinguishable trunk, main and secondary branches, the theory of partial differential equations has the appearance of a bush with very complex structure. Its roots as well as its flowers are often related to the physical world, and it is fertilized by ideas and techniques borrowed from virtually every other area of mathematics. Evolutionary partial differential equations made their first appearance in the 18th century, in the endeavor to understand the motion of fluids and other continuous media. It is remarkable that this research program is still ongoing and many fundamental questions remain unanswered. Beyond this area, however, evolutionary partial differential equations have become ubiquitous, as they seem to govern the dynamics of any physical, chemical biological, ecological or economic system whose state is described by spatially dependent variables. The active research effort over the span of two centuries, combined with the wide variety of physical phenomena that had to be explained, has resulted in an enormous body of literature. Any attempt to produce a comprehensive survey would be futile. The aim here is to collect review articles, written by leading experts, which will highlight the present and expected future directions of development of the field. The emphasis will be on nonlinear equations, which pose the most challenging problems today. The various articles will offer the reader the opportunity to compare and contrast the behavior of hyperbolic and parabolic equations. Hyperbolic equations are typically associated with media exhibiting “elastic” response, and encompass the notoriously difficult class of “hyperbolic conservation laws”. Parabolic equations are associated with various diffusive mechanisms. An extremely important and challenging example is the Navier–Stokes equation. Volume I of this Handbook will focus on the abstract theory of evolutionary equations, addressing questions of existence, uniqueness and other general qualitative properties of solutions. Future volumes will consider more concrete problems relating to specific applications. Our hope is that the handbook will provide a panorama of this amazingly complex and rapidly developing branch of mathematics.