ABSTRACT
This chapter examines solutions (global in time) to systems describing the motion of both incompressible liquids and compressible isothermal gases in a bounded domain. The aim is to find global solutions to equations of motion for models of both incompressible and compressible fluids. The chapter tries to answer the questions: (1) For which p do measure-valued solutions exist? (2) For which p are measure-valued solutions Dirac ones?—or equivalently?—For which p do weak solutions exist? and (3) For which p, if any, do weak solutions have some qualitative properties such as higher regularity, uniqueness, etc? It discusses Korn’s inequality, and demonstrates the existence of a measure-valued solution to the problem (NS)p with the aim of illustrating some basic techniques and methods used in the evolution theory of incompressible fluids.
