# 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?

## 1 Answer

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.

It is interesting to notice that the opposite of Brenner's condition is that the characteristic cone of the differential operator has maximal curvature, hence the Cauchy problem admits a Strichartz-like estimate. You can read the discussion in the 1st chapter of my book co-authored with S. Benzoni-Gavage.

• This is very interesting. Thank you. Could you add some references for the claims regarding the one-dimensional case? – Riku Apr 22 at 20:56
• @Riku: the one dimensional case is described in A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, Oxford, 2000. The result of Rauch referred to is Commun. Math. Phys. 106, 481--484 (1986). – Willie Wong Apr 22 at 21:15
• @WillieWong Isn't book of Bressan about conservation laws and not about wave equations? – Riku Apr 22 at 21:16
• @WillieWong In fact, why is the answer talking about entropy solutions? – Riku Apr 22 at 21:17
• @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 at 21:18