Narrow Results

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small $H^s$-initial conditions (with $s$ sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition.

We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy. As an application we also show a conformal energy estimate with vector fields applied to the solution as well as a global $L^{\infty }$ decay bound in terms of a weighted norm on initial data. For solutions to the wave equation in these dynamical backgrounds, our results reduce the problem of establishing the classical pointwise decay rate $t^{-3/2}$ in the interior and $t^{-1}$ along outgoing null cones to simply proving that local energy decay holds.