In the book of Evans the transport equation, $$\frac{d}{dt} u + b\cdot \nabla u = 0, \quad u(t=0)=u_0,$$ is solved by the method of charateristics for $b$ and $u_0$ smooth enogh (in terms of $\mathcal{C}^k$ for some $k$).
I am not an expert for hyperbolic equations and I see this equation as a prototyp of hyperbolic equations of first order and I thought that there are some results of this equation in Sobolev spaces. But I haven't found some (expect for strong solutions or the result of DiPerna and Lions; see below).
To be more precise: I wonder if there is any result of the type: If $b$ and $u_0$ are contained in some Sobolev spaces, then $u\in L^p(W^{k,q})=L^p(0,T;W^{k,p})$ for $k\geq 1$ and some $p,q$? I know the paper of DiPerna and Lions from 1989, where develop a theory of this equation in Sobolev spaces, but they don't study if the solution $u$ has a (weak) derivative (or even more).
Does anyone knows an article for this equation or (if not) a good reference (book or survey article) for beginners in hyperbolic equations?
Thanks a lot!
Edit: 1) $b$ can be assumed as divergence free, $\nabla \cdot b=0$. 2) Higher regularity (in terms of derivates for $u$) can be established by the method, Willi Wong mentioned in his comment. This question is okay now. Does anyone has a reference for a beginng guide for hyperbolic equations or a survey article with further archiements than DiPerna and Lions?