  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].
