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?

conserves every spatial $L^q$ norm, which means that it cannot have the decay in time necessary for $L^p$ in time except with $p = \infty$. $\endgroup$ – Willie Wong Jan 27 '11 at 17:24ODE, transport theory, and Sobolev spacesin Inventione). Consider equation 11. Commute your transport equation with a derivative, you have $$ \partial_t (\partial u) + b\cdot \nabla (\partial u) + (\partial b)\cdot \nabla u = 0 $$ so $\partial u$ solves a system of the form equation 11. In general, commuting equation 11 with $\partial^k$ shows that the tower $(u, \partial u, \partial^2u, \ldots,\partial^ku)$ solves a transport equation also of form 11. $\endgroup$ – Willie Wong Jan 27 '11 at 17:37