Well-posedness of the IBVPs for the 1D viscous gas equations

P. 1–73.

Zlotnik Alexander

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.

Keywords: viscous compressible Navier-Stokes equations, well-posedness, initial-boundary value problems, viscous barotropic gas, viscous heat-conducting gas, one-dimensional equations, Lagrangian mass coordinates, weak solutions

In book

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Springer, 2017.