Long Time Boundedness of Planar Jump Discontinuities for Homogeneous Hyperbolic Systems

TL;DR

This paper investigates the boundedness over time of solutions to certain hyperbolic PDEs with jump discontinuities, identifying conditions under which solutions remain uniformly bounded despite potential gradient jumps.

Contribution

It establishes conditions related to the characteristic variety's geometry that ensure long-time boundedness of solutions with discontinuities for homogeneous hyperbolic systems.

Findings

Solutions are uniformly bounded when the characteristic variety's Hessian rank is zero or maximal.

Gradient jumps across characteristic hyperplanes do not grow unbounded over time under specified conditions.

The paper clarifies the relationship between the geometry of the characteristic variety and solution boundedness.

Abstract

Suppose that $L(\partial_t,\partial_x)$ is a homogeneous constant coefficient strongly hyperbolic partial differential operator on ${\mathbb R}^{1+d}$ and $H$ is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of $H$, the characteristic variety of $L$ is the graph of a real analytic function $\tau(\xi)$ with ${\rm rank}\,\tau_{\xi\xi}$ identically equal to zero or the maximal possible value $d-1$. Suppose that the source function $f$ is compactly supported in $t\ge 0$ and piecewise smooth with singularities only on $H$. Then the solution of $Lu=f$ with $u=0$ for $t<0$ is uniformly bounded on ${\mathbb R}^{1+d}$. Typically when ${\rm rank}\,\tau_{\xi\xi}\ne 0$ on the conormal variety, the sup norm of the the jump in the gradient of $u$ across $H$ grows linearly with $t$.