Abstract
In this paper, we extend our previous result from Levy (2016). We prove that transport equations with rough coefficients do possess a uniqueness property, even in the presence of viscosity. Our method relies strongly on duality and bears a strong resemblance to the well-known DiPerna-Lions theory first developed by DiPerna and Lions (1989). This uniqueness result allows us to reprove the celebrated theorem of Serrin (1962) in a novel way. As a byproduct of the techniques, we derive an L1 bound for the vorticity in terms of a critical Lebesgue norm of the velocity field. We also show that the zero solution is unique for the 2D Euler equations on the torus under a mild integrability assumption.
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