Uniqueness for some Leray–Hopf solutions to the Navier–Stokes equations

We exhibit simple sufficient conditions which give weak–strong uniqueness for the 3D Navier–Stokes equations. The main tools are trilinear estimates and energy inequalities. We then apply our result to the framework of Lorentz, Morrey and Besov over Morrey spaces so as to get new weak–strong uniqueness classes and so uniqueness classes for solutions in the Leray–Hopf class. In the last section, we give a uniqueness and regularity result. We obtain new uniqueness classes for solutions in the Leray–Hopf class without energy inequalities but sufficiently regular.