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?

  • $\begingroup$ By $L^p(W^{k,q})$ do you mean $L^p$ in time and $W^{k,q}$ in space? In general unless you localize in time you certainly cannot get something that strong, just by using regularity of $b$. You will also need algebraic structure. For example, in the case where $\nabla\cdot b = 0$, you have that the transport equation 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 17:24
  • $\begingroup$ @Willie Wong: Yes, I mean $L^p(W^{k,p}):=L^p(0,T;W^{k,p})$ for a bounded time inverval $[0,T]$. I'm much interested in the case where $b$ is divergence free, but informations about other cases are okay, too. $\endgroup$ – Markus Klein Jan 27 '11 at 17:31
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    $\begingroup$ Actually... I think your question is already answered in DiPerna and Lion's paper (I assume you meant ODE, transport theory, and Sobolev spaces in 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
  • $\begingroup$ So apply the result there to the tower being in $L^q$ gives you a result for $u$ in $W^{k,q}$, which also requires upgrading the Sobolev space assumption on $b$ and $c$ to contain more derivatives. $\endgroup$ – Willie Wong Jan 27 '11 at 17:38
  • $\begingroup$ @Wille Wong: Thanks for your answer. I think, this is the point for higher regularity. Thanks! Do you (or others) know if there is a further development for this equation beyond the paper of DiPerna and Lions? $\endgroup$ – Markus Klein Jan 28 '11 at 14:10

You can start with an interesteing extension - Ambrosio 2004 showed existence, uniqueness and stability results for $b\in BV$ and ${\rm div}_x b$ absolutely continuous with respect to Lebesgue measure.

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