Weak uniqueness and the Serrin criterion for the Navier-Stokes equations

This talk is divided into two parts. In the first one, we begin by proving a linear uniqueness result for weak solutions of transport-diffusion equations possessing some integrability. This first theorem follows the ideas of the celebrated DiPerna-Lions theory, whose main lines will be recalled. In a second part, we use this uniqueness result - along with variations thereof - to show the smoothness of a Leray solution of the (incompressible, homogeneous) Navier-Stokes equations of which only one component is assumed to satisfy a regularity assumption at the scaling of the equations.