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

Abstract

In this paper, we consider the non-uniqueness of transport equation on the torus ${\mathbb{T}}^d$, with density $\rho \in L_t^s{L}_x^p$ and divergence-free vector field $\mathbf{u}\in L_t^{s^{\prime }}{L}_x^{p^{\prime }}\cap L_t^{\tilde{s}}{W}_x^{1,\tilde{p}}$. We prove that the non-uniqueness holds for $\frac{1}{p}+\frac{{\tilde{s}}^{\prime }}{s\tilde{p}}>1+\frac{1}{d-1}$, with $d\ge 2$ and $s,p,\tilde{p}\in [1,\infty )$, $1\le \tilde{s}<s^{\prime }$. The result can be extended to the transport-diffusion equation with diffusion operator of order $k$ in the class $\rho \in L_t^s{L}_x^p\cap L_t^{\bar{s}}{C}_x^{\bar{m}}$, $\mathbf{u}\in L_t^{s^{\prime }}{L}_x^{p^{\prime }}\cap L_t^{\tilde{s}}{W}_x^{1,\tilde{p}}$, under some conditions on $\bar{s},\bar{m},k$. In particular, when $\tilde{s}=1$, the additional condition is $\bar{m}<\frac{s}{\bar{s}}-1$, $k<\frac{s}{s^{\prime }}+1$. 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-Székelyhidi [Ann. PDE, 2018; Calc. Var. Partial Differ. Equ., 2019; Annales de l'Institut Henri Poincaré C, Analyse non linèaire, 2020] and Cheskidov-Luo [Ann. PDE, 2021; arXiv, 2022 (forthcoming in Anal. PDE, 2023)].

arXiv admin note: text overlap with arXiv:2308.01506