Shock Formation in Small-Data Solutions to $3D$ Quasilinear Wave Equations

Abstract

In his 2007 monograph, D. Christodoulou proved a breakthrough result giving a detailed description of the formation of shocks in solutions to the relativistic Euler equations in three spatial dimensions. He assumed that the data have small $H^N$ norm, where $N$ is a sufficiently large integer. To deduce the shock formation, he also assumed that the data verify a signed integral inequality. In the present monograph, we extend Christodoulou's framework and use it to prove that shock singularities often develop in initially small, regular solutions to two important classes of quasilinear wave equations in three spatial dimensions. Our work also generalizes and unifies earlier work on singularity formation initiated by F. John in the 1970's and continued by L. H\"ormander, S. Alinhac, and many others. Specifically, we study $\mbox{i)}$ covariant scalar wave equations of the form $\square_{g(\Psi)} \Psi = 0$ and $\mbox{ii)}$ non-covariant scalar wave equations of the form $(h^{-1})^{\alpha \beta}(\partial \Phi) \partial_{\alpha} \partial_{\beta} \Phi = 0.$ Our main result shows that whenever the nonlinear terms fail Klainerman's classic null condition, shocks develop in solutions arising from an open set of small data. Hence, within the classes $\mbox{i)}$ and $\mbox{ii)}$, our work can be viewed as a sharp converse to the well-known result of Christodoulou and Klainerman, which showed that when the classic null condition is verified, small-data global existence holds.