On the non-uniqueness of transport equation: the quantitative relationship between temporal and spatial regularity

In this paper, we consider the non-uniqueness of transport equation on the torus $\mathbb{T}^d$, with density $\rho\in L^{s}_tL_x^{p}$ and divergence-free vector field $\boldsymbol{u}\in L^{s'}_tL_x^{p'}\cap L^{\tilde{s}}_tW_x^{1,\tilde{p}}$. We prove that the non-uniqueness holds for $\frac{1}{p}+\frac{\tilde{s}'}{s\tilde{p}}>1+\frac{1}{d-1}$, with $d\ge 2$ and $s,p,\tilde{p}\in[1,\infty)$, $1\le\tilde{s}<s'$. The result can be extended to the transport-diffusion equation with diffusion operator of order $k$ in the class $\rho\in L^{s}_tL_x^{p}\cap L_t^{\bar{s}}C_x^{\bar{m}}$, $\boldsymbol{u}\in L^{s'}_tL_x^{p'}\cap L^{\tilde{s}}_tW_x^{1,\tilde{p}}$, under some conditions on $\bar{s},\bar{m},k$. In particular, when $\tilde{s}=1$, the additional condition is $\bar{m}<\frac{s}{\bar{s}}-1$, $k<\frac{s}{s'}+1$. These results can be considered as quantitative versions of Cheskidov and Luo's [Ann. PDE, 2021]. The main tool is the convex integration developed by Modena-Sattig-Sz\'ekelyhidi [Ann. PDE, 2018; Calc. Var. Partial Differ. Equ., 2019; Annales de l'Institut Henri Poincar\'e C, Analyse non lin\`eaire, 2020] and Cheskidov-Luo [Ann. PDE, 2021; arXiv, 2022 (forthcoming in Anal. PDE, 2023)].