Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity

Abstract

We consider the linear transport equations driven by an incompressible flow in dimensions \(d\ge 3\). For divergence-free vector fields \(u \in L^1_t W^{1,q}\), the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class \(L^\infty _t L^p\) when \(\frac{1}{p} + \frac{1}{q} \le 1\). For such vector fields, we show that in the regime \(\frac{1}{p} + \frac{1}{q} > 1\), weak solutions are not unique in the class \( L^1_t L^p\). One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.