Energy splitting for solutions of multi-dimensional isotropic symmetric hyperbolic systems

Abstract

This note, submitted in honor of Walter Strauss’ 70th birthday, offers a new look at the elegant paper by Costa and Strauss entitled Energy Splitting, [3]. In it, they studied finite energy solutions to linear, constant coefficient, isotropic, symmetric hyperbolic systems. They showed that solutions separate, in L2, into a superposition of outgoing plane waves, using a decomposition based on the Radon transform. Our approach to energy splitting is based on the local energy decay result obtained by the author with B. Thomases in [9], for rotationally and scaling invariant isotropic systems. Inspired by Costa and Strauss, here we replace rotational invariance by the isospectral condition of [3]. This involves a nonlocal modification of the rotational vector fields. However, the final result is entirely local. Solutions are decomposed locally according to the spectral projections of the symbol, and individual wave families concentrate near characteristic cones. After a few preliminaries, we state the main result and compare it with [3]. The proof is entirely self-contained. To conclude, we illustrate with three examples. Energy splitting is closely related to the phenomenon of energy equipartition. In the context of symmetric hyperbolic systems with constant coefficients, there have been numerous works on these topics, among them [2, 4, 10, 11].