Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Elia Bruè, Maria Colombo, Anuj Kumar

Given a divergence-free vector field $\mathbf{u}\in L_t^{\infty }{W}_x^{1,p}({\mathbb{R}}^d)$ and a nonnegative initial datum ${\rho }_0\in L^r$, the celebrated DiPerna--Lions theory established…