Non-uniqueness of the transport equation at high spacial integrability

Abstract

In this paper, we show the non-uniqueness of the weak solution in the class $\rho \in L_t^s{L}_x^p$ for the transport equation driven by a divergence-free vector field $\mathbf{u}\in L_t^{\tilde{s}}{W}_x^{1,q}\cap L_t^{s^{\prime }}{L}_x^{p^{\prime }}$ happens in the range $1/p+1/q>1-\frac{p-1}{4(p+1)p}$ with some $\tilde{s}>1$, as long as $1\le s<\infty$, $p>1$. As a corollary, $L^{\infty }$ in time of the density $\rho$ is critical in some sense for the uniqueness of weak solution. Our proof is based on the convex integration method developed in [Modena and Sattig, 2020, Ann. Inst. H. Poincaré C Anal. Non Linéaire], [Cheskidov and Luo, 2021, Ann. PDE].