Handbook of Mathematical Analysis in Mechanics of Viscous Fluids
Mathematics has always played a key role for researches in fluid mechanics. The purpose of this handbook is to give an overview of items that are key to handling problems in fluid mechanics. Since the field of fluid mechanics is huge, it is almost impossible to cover many topics. In this handbook, we focus on mathematical analysis on viscous Newtonian fluid. The first part is devoted to mathematical analysis on incompressible fluids while part 2 is devoted to compressible fluids.
The inhomogeneous initial-boundary value problems (IBVPs) are posed for the Navier-Stokes systems of equations describing the viscous barotropic and heat-conducting gas 1D flow in the Lagrangian mass coordinates. Weak solutions are studied without any restrictions on the magnitude of norms of data. Assumptions on the data are genuinely general, in particular, the initial data are taken from the Lebesgue spaces, the contact problems for different gases are covered, etc. Both the global in time existence of the weak solutions as well as their uniqueness and Lipschitz continuous dependence on data are proved thus ensuring the well-posedness of the IBVPs. The regularity issue is studied as well.