Conormal and piecewise smooth solutions to quasilinear wave equations

Abstract:

In this paper, we show first that if a solution $u$ of the equation ${P_2}(t,x,u,Du,D)u = f(t,x,u,Du)$, where ${P_2}(t,x,u,Du,D)$ is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface $\Sigma$ of ${P_2}$ in the past and $\Sigma$ is smooth in the past, then $\Sigma$ is smooth and $u$ is conormal with respect to $\Sigma$ for all time. Second, let ${\Sigma _0}$ and ${\Sigma _1}$ be characteristic hypersurfaces of ${P_2}$ which intersect transversally and let $\Gamma = {\Sigma _0} \cap {\Sigma _1}$. If ${\Sigma _0}$ and ${\Sigma _1}$ are smooth in the past and $u$ is conormal with repect to $\{ {\Sigma _0},{\Sigma _1}\}$ in the past, then $\Gamma$ is smooth, and $u$ is conormal with respect to $\{ {\Sigma _0},{\Sigma _1}\}$ locally in time outside of $\Gamma$, even though ${\Sigma _0}$ and ${\Sigma _1}$ are no longer necessarily smooth across $\Gamma$. Finally, we show that if $u(0,x)$ and ${\partial _t}u(0,x)$ are in an appropriate Sobolev space and are piecewise smooth outside of $\Gamma$, then $u$ is piecewise smooth locally in time outside of ${\Sigma _0} \cup {\Sigma _1}$.