Convex integration solutions to the transport equation with full dimensional concentration

Abstract

We construct infinitely many incompressible Sobolev vector fields u \in C_{t}W_{x}^{1,\tilde p} on the periodic domain \mathbb{T}^{d} for which uniqueness of solutions to the transport equation fails in the class of densities \rho \in C_{t}L_{x}^{p} , provided 1/ p + 1/ \tilde p> 1 + 1/ d . The same result applies to the transport-diffusion equation, if, in addition, p^{′} < d .

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