Convex integration method and non-uniqueness of weak solutions for viscous fluids (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

We review our recent results concerning the non-uniqueness of weak solutions to viscous fluid dynamics, based on the technique of convex integration. Concerning the hyperdissipative incompressible Navier-Stokes and MHD equations, we proved the sharp non-uniqueness of weak solutions in supercritical regimes with respect to the Ladyženskaja-Prodi-Serrin (LPS) criteria. For the hypo-viscous compressible Naver-Stokes equations, we proved that there exist infinitely many weak solutions with the same initial data, which provides the first non-uniqueness result of weak solutions to viscous compressible fluid.