# BV functions and wave equation

What is the role played by BV functions in the study of (possibly nonlinear) wave equations?

I think that one would need assume small initial data in $$L^1$$ or $$H^1$$ to get a well-posedness result (is that correct?).

Has the case of initial data in BV been studied?

The answer to this question depends a lot on the space dimension $$n$$. It is true that if $$n=1$$, the Cauchy problem has been studied with data in either $$L^\infty(R)$$ or $$BV(R)$$. For superlinear wave equation, every $$L^\infty$$-data yields at least one bounded global-in-time "entropy" solution. This is done by Compensated Compactness (DiPerna 1983). However, nobody knows whether this solution is unique. The $$BV$$ space is better suited in some sense, because Bressan was able to prove uniqueness of the entropy solution ; however, existence is obtained only if the initial data $$u_0$$ is not too large, in the sense that $$\|u_0\|_\infty TV(u_0)<\delta$$ for some absolute finite constant $$\delta>0$$. In some sense, the Cauchy problem is well-posed in $$BV$$, in a neighbourhood of constant data.
In several space dimensions, Rauch remarked that you should forget the $$BV$$ space. The Cauchy problem cannot be well-posed in this topology. The reason is that the Cauchy problem for the linear wave equation itself is ill-posed in $$BV$$. This is a consequence of a theorem by Brenner in the 60's, which tells that this Cauchy problem is ill-posed in every $$L^p$$-space, except for the Hilbert case $$p=2$$. Brenner's theorem is a bit more complete. It says that for a first-order hyperbolic system $$\partial_tf+\sum_{j=1}^nA_j\partial_jf=0,$$ the Cauchy problem is well-posed in some $$L^p$$ with $$p\ne2$$ if, and only if the matrices $$A_j$$ commutte to each other. This very strong condition amounts to saying that the system can be rewritten as a list of decoupled transport equations.
• @Riku: a wave equation is a particular form of a conservation law. For example, if you write $-\partial_t^2 \phi + \partial^2_x\phi = 0$ for the linear wave equation, and set $\psi_1 = \partial_t \phi$ and $\psi_2 = \partial_x \phi$, then the wave equation is equivalent to the conservation laws $\partial_t \psi_1 - \partial_x \psi_2 = 0$ coupled to $\partial_t \psi_2 - \partial_x \psi_1 = 0$. – Willie Wong Apr 22 '19 at 21:18