A review of basic theoretical results concerning the Navier-Stokes and other similar equations

Abstract

These notes are devoted to provide an introductory approach to the Navier-Stokes and some other related equations. Most concepts and arguments recalled below are very general and we believe that this presentation can be of help for the theoretical analysis of many PDE’s arising from Sciences and Engineering. First, we recall the Navier-Stokes equations, we explain the meaning of the variables and data and we state some technical results needed for our study. Then, we state and give the proofs of some basic existence, uniqueness and regularity results. In the proof of existence, we apply usual compactness arguments to a family of Galerkin approximations. We also discuss briefly some of the main open problems arising in the three-dimensional case. In a final section, we review briefly the state of the art for other similar equations and we indicate some related open questions.