Property of a transport equation

Question

Let $\Omega\subset\mathbb{R}^N, N=2$ or $N=3$ be a bounded smooth domain, $T>0$ and $y=y(x,t)\in L^{\infty}(\Omega\times (0,T))^N$ is a given vector function such that $\nabla\cdot y=0$. Consider the tranform equation

$$\dfrac{\partial u}{\partial t}-\nabla\cdot (yu)=f(x,t), \mbox{for }(x,t)\in \Omega\times (0,T);\\ u_{t=0}=u_0, \ x\in \Omega.$$ How about the existence and regularity of solution of this transport equation?

For example, we need $u\in L^2(0,T;H_0^1(\Omega)),$ then what are $f$ and $u_0$?

Answer 1

I am afraid the transport vector do not have enough regularity. look at these papers Transport equation and Cauchy problem for BV vector fields by Luigi Ambrosio and ODEs,transport theory,and Sobolev spaces Diperna&P.Lions Also their results do not give so strong results,only $L^{\infty}(0,T,L^p)$ when the transport vector is in certain Sobolev spaces.