Non-uniqueness in law of three-dimensional Navier-Stokes equations diffused via a fractional Laplacian with power less than one half

Abstract: We review recent developments on the technique of convex integration that has led to proofs of the non-uniqueness of PDEs in fluid mechanics such as the Euler equations, Navier-Stokes equations, and Boussinesq system, in both deterministic and stochastic cases. In the stochastic case we will focus on the non-uniqueness in law of the Navier-Stokes equations diffused via a fractional Laplacian (power equals 1 gives the classical Navier-Stokes equations): dimension 3 with power between 13/20 and 5/4, as well as power between 0 and 1/2; dimension 2 with power between 0 and 1. We also discuss analogous results concerning non-uniqueness in law of the Boussinesq system and conclude by listing several open problems.