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, $p^{\prime }<d$.