TABLE OF CONTENTS The aim of this book is to develop a theory of initial-boundary value problems for linear and nonlinear partial differential equations. There are already many books available, and we shall list some of them after the introduction. However, the area is vast, and any one book can only treat certain aspects of the theory. Our choice of material is very much influenced by the availability of fast computers. They have made it possible to solve rather complex problems for which the classical theory of second-order equations is not adequate. Existence and regularity questions play a fundamental r le in computations because the resolution required depends on the smoothness of the solution, and there is always the danger that one tries to compute things which do not exist. Another fundamental question concerns admissible boundary conditions, which we shall discuss in great detail. In computations the boundary conditions cause most of the problems.
We believe that this book can fill a gap between elementary and rather abstract books. To illustrate our theory, we have chosen the compressible and incompressible Navier-Stokes (N-S) equations, which describe fluid flows ranging from large scale atmospheric motions to the lubrication of ball bearings. The choice was dictated by the desire to find a system which is so rich in phenomena that the whole power of the mathematical theory is needed to discuss existence, smoothness and boundary conditions. We hasten to add, however, that we only scratch the surface of the diversity in which its solutions can behave.
