A quasilinear wave with a supersonic shock in a weak solution interrupting the classical development

TL;DR

This paper constructs and analyzes a model quasilinear wave equation in 1+1 dimensions, demonstrating the existence, uniqueness, and structure of classical and weak shock solutions, including a novel example of a maximal globally hyperbolic development with shocks.

Contribution

It provides the first example of a provably unique maximal globally hyperbolic development for a shock-forming quasilinear wave equation, illustrating phenomena of shocks and singularities in a simple model.

Findings

Existence of classical solutions that blow up in finite time.

Construction of a unique maximal globally hyperbolic development (MGHD) with a well-described boundary.

Existence of a unique weak entropy solution with a shock curve, matching the classical solution before shock formation.

Abstract

We study the Cauchy problem for classical and weak shock-forming solutions to a model quasilinear wave equation in $1+1$ dimensions arising from a convenient choice of $C^{\infty}$ initial data, which allows us to solve the equation using elementary arguments. The simplicity of our model allows us to succinctly illustrate various phenomena of geometric and analytic significance tied to shocks, which we view as a prototype for phenomena that can occur in more general quasilinear hyperbolic PDE solutions. Our Cauchy problem admits a classical solution that blows up in finite time. The classical solution is defined in a largest possible globally hyperbolic region called a maximal globally hyperbolic development (MGHD), and its properties are tied to the intrinsic Lorentzian geometry of the equation and solution. The boundary of the MGHD contains an initial singularity, a singular boundary along which the solution's second derivatives blow up (the solution and its first derivatives remain bounded), and a Cauchy horizon. Our main results provide the first example of a provably unique MGHD for a shock-forming quasilinear wave equation solution; it is provably unique because its boundary has a favorable global structure that we precisely describe. We also prove that for the same $C^{\infty}$ initial data, the Cauchy problem admits a second kind of solution: a unique global weak entropy solution that has a shock curve separating two smooth regions. Of particular interest is our proof that the classical and weak solutions agree before the shock but differ in a region to the future of the first singularity where both solutions are defined.