Blow-up and uniqueness of Leray-Hopf solutions to forced Navier-Stokes equations

Abstract

We prove the existence of forces in $L^2_{\rm loc}(\R_+;L^{6/5}(\R^3))$ and smooth initial data such that the associated Leray-Hopf solution of the 3D Navier-Stokes equations is unique, global-in-time, satisfies the energy equality, and has a blow-up instant. The same statement is obtained for forces in $L^{5/4}_{\rm loc}(\R_+;L^2(\R^3))$. These results are extended to countably many blow-up instants. By strengthening the integrability assumptions on the force we also prove blow-up of Leray-Hopf solutions in the original sense intended by Leray. Our method also allows us to construct examples of blow-up for the forced 3D Euler equations.