Grant Details

The present proposal addresses a collection of problems concerning the mathematical analysis of partial differential equations arising in compressible fluid mechanics. In particular, the proposal concerns the classical systems of isentropic Euler and Navier-Stokes equations, and aims at establishing new and fundamental results which will have a major impact in the community of Applied Partial Differential Equations. The specific objectives of the proposal concern the existence of weak solutions for the compressible Navier-Stokes equations with density dependent coefficients in configurations with vacuum, the existence of dissipative Hölder solutions to the compressible isentropic Euler equations and the existence of pathological non unique weak solutions for the Navier-Stokes equations, which will extend to the compressible world groundbreaking results recently obtained in the context of incompressible fluids. Among the main tools which will be developed in the present proposal there will be a new designed convex integration scheme for compressible fluids and a suitable adaptation of the theory of renormalized solutions for transport and continuity equations with non-smooth vector fields in the context of compressible fluids.