Program

Introduction to Compressible Navier-Stokes. Derivation of the model via physical laws (Conservation of mass and momentum). Energy inequality for the system. Definition of the weak solutions. Concept of renormalized solutions. Heuristically explain the main idea behind the existence proof: i. compactness of weak solutions, ii. Estimates from energy, iii. Improved estimates of density, iv. Limit passage, v. Effective viscous flux, vi. Strong Convergence of density.

Construction of approximate solutions: three levels of approximations. Boundedness of the density for approximate levels.

Existence of global solution in the Galerkin level. Derive several estimates independent of the dimension of Galerkin approximations.

Existence of complete system with dissipation in the continuity equation and artificial pressure in the momentum equation.

The vanishing viscosity limit. Passing to the limit in the artificial pressure.

Local-in-time existence of strong solutions: i. Lagrangian change of variables, ii. Analysis of linear problem, iii. Estimates of the nonlinear terms, iv. Fixed point argument.

Global-in-time existence with small data: i. A priori estimates, ii. Extension of solution.

Notion of Relative energy. Finite energy weak solutions satisfy the relative energy inequality.

Weak-strong uniqueness on bounded domains.