On well-posedness of systems arising from fluid dynamics

Well-posedness of systems describing the motion of compressible fluids in the class of strong and weak solutions represents one of the most challenging problems of the modern theory of PDEs. In the first part of the talk, we are going to define and construct a larger class of solutions, the measure-valued and dissipative ones, which unable us to handle the problem of existence for large times and large initial data; we will also discuss the possible advantages of considering this weaker notion of solution when solving some related problems arising from fluid dynamics. In the second part of the talk, we are going to show that, by performing a suitable selection, it is possible to select one "good" solution satisfying the semigroup property, even in the context when the system lacks uniqueness. After showing this procedure in an abstract setting, we will apply it to specific systems, such as the compressible Euler and Navier-Stokes ones.