On well-posedness of weak solutions for quasilinear evolutionary PDEs

The standard Hadamard's well-posedness theory has been established for linear systems, or for smooth solutions of nonlinear systems. For weak solutions of nonlinear systems, the usual calculus and functional analytic methods do not yield well-posedness theory. The difficulty is due in large part to the lack of understanding of the structure of weak solutions. In fact, there are several striking ill-posed examples for incompressible Euler and Navier-Stokes equations, and also for compressible Euler equations. On the other hand, there is the well-known well-posedness theory for the system of hyperbolic conservation laws, starting with the existence theory of James Glimm. We will survey the development of these. We will also explain the recent well-posedness theory for the compressible Navier-Stokes equation done in collaboration with Shih-Hsien Yu.