On the formation of shocks for quasilinear wave equations

Abstract

The paper is devoted to the study of shock formation of the 3-dimensional quasilinear wave equation

where $G^{\prime \prime }(0)$ is a non-zero constant. We will exhibit a family of smooth initial data and show that the foliation of the incoming characteristic hypersurfaces collapses. Similar to 1-dimensional conservational laws, we refer this specific type breakdown of smooth solutions as shock formation. Since $(\star )$ satisfies the classical null condition, it admits global smooth solutions for small data. Therefore, we will work with large data (in energy norm). Moreover, no symmetry condition is imposed on the initial datum. We emphasize the geometric perspectives of shock formation in the proof. More specifically, the key idea is to study the interplay between the following two objects: (1) the energy estimates of the linearized equations of $(\star )$; (2) the differential geometry of the Lorentzian metric $g=-\frac{1}{(1+3G^{\prime \prime }(0) (\partial _t\phi )^2)} d t^2+dx_1^2+dx_2^2+dx_3^2$. Indeed, the study of the characteristic hypersurfaces (implies shock formation) is the study of the null hypersurfaces of g. The techniques in the proof are inspired by the work (Christodoulou in The Formation of Shocks in 3-Dimensional Fluids. Monographs in Mathematics, European Mathematical Society, 2007) in which the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established. We also use the short pulse method which is introduced in the study of formation of black holes in general relativity in Christodoulou (The Formation of Black Holes in General Relativity. Monographs in Mathematics, European Mathematical Society, 2009) and generalized in Klainerman and Rodnianski (Acta Math 208(2):211–333, 2012).