Let $\Omega\subset\mathbb{R}^N, N=2$ or $N=3$, be a smooth bounded domain and $T>0$. Given a vector function $h: \Omega\times (0,T)\to \mathbb{R}^N$ such that $\nabla\cdot h=0$. For any function $f$, consider the transport equation \begin{equation} \begin{cases} \dfrac{\partial u}{\partial t}+h\cdot\nabla u=f(x,t), (x,t)\in \Omega\times (0,T)\\ u_{|t=0}=u_0(x), x\in \Omega. \end{cases}\end{equation} I need a sufficient assumption on functions $h$ and $f$ and the inital data $u_0$ such that the above problem has at least one weak solution $u\in L^{\infty}(0,T;W^{1,\infty}(\Omega))\cap C([0,T];L^2(\Omega))$.
For example, if we take $h\in L^{\infty}(0,T;W^{1,\infty}(\Omega)^{N}), f\in L^{\infty}(0,T;W^{1,\infty}(\Omega))$ and $u_0\in W^{1,\infty}(\Omega)$, then Does that $u\in L^{\infty}(0,T;W^{1,\infty}(\Omega))\cap C([0,T];L^2(\Omega))$?
Would someone help me, please?