Non-uniqueness of the transport equation at high spacial integrability

In this paper, we show the non-uniqueness of the weak solution in the class $\rho\in L^{s}_tL^p_x$ for the transport equation driven by a divergence-free vector field $\boldsymbol{u}\in L^{\tilde{s}}_tW^{1,q}_x\cap L_t^{s'}L_x^{p'}$ 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. Poincar\'e C Anal. Non Lin\'eaire], [Cheskidov and Luo, 2021, Ann. PDE].