Handbook of Differential Equations Evolutionary Equations Vol IV

Abstract

The present, fourth volume in the series Evolutionary Equations of the Handbook of Differential Equations develops further the program initiated in the past three volumes, namely to provide a panorama of this amazingly rich field, whose roots and fruits are related to the physical world while its flowers belong to the world of mathematics. With an eye towards retaining the proper balance between basic theory and its applications, we are including here review articles by leading experts on the following topics. Chapter 1, by D. Chae, deals with equations related to the Euler equations for incompressible fluids, and examines the development of singularities in finite time. The recent development in the mathematical theory of the compressible Navier–Stokes equations is addressed in Chapter 2 by E. Feireisl. In Chapter 3, A. Miranville and S. Zelik discuss the large time behavior of solutions of dissipative partial differential equations, in bounded or unbounded domains, and establish, in particular, the existence of global and exponential attractors. The aim of Chapter 4, by A. Novick-Cohen, is to present recent results in the theory of the Cahn–Hilliard equation as well as related problems. The problem of existence, regularity and stability of solutions to systems of evolutionary equations governing the flow of viscoelastic fluids is the focus of Chapter 5, by M. Renardy. The following Chapter 6, by L. Simon, is devoted to the application of the theory of monotone operators to parabolic and functional-parabolic equations or systems thereof. In Chapter 7, by A. Vasseur, the recent results in hydrodynamic limits, especially those corresponding to hyperbolic scaling, are addressed. Chapter 8, by A. Visintin, gives a detailed introduction into the modeling of phenomena which can be described by the Stefan-type problems together with analysis of their weak formulation. Finally, A. Wazwaz’s Chapter 9 deals with the Korteweg–deVries equation and some of its modifications and describes various methods for constructing solutions. We are indebted to the authors, for their valuable contributions, to the referees, for their helpful comments, and to the editors and staff of Elsevier, for their assistance