Nontrivial global solutions to some quasilinear wave equations in three space dimensions

TL;DR

This paper develops a method to construct nontrivial global solutions for certain quasilinear wave equations in three dimensions, applying it to classical counterexamples and Euler equations, thereby expanding understanding of their solution space.

Contribution

It introduces a conditional framework for constructing global solutions to quasilinear wave equations and explicitly applies it to important equations like John's counterexamples and Euler equations.

Findings

Existence of large families of global solutions to specific quasilinear wave equations.

Construction of explicit global solutions satisfying key pointwise bounds.

Validation of the conditional approach for important classes of wave equations.

Abstract

In this paper, we seek to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result on the construction of nontrivial global solutions to a general system of quasilinear wave equations. Assuming that a global solution to the geometric reduced system exists and satisfies several well-chosen pointwise estimates, we find a matching exact global solution to the original wave equations. Such a conditional result is then applied to two types of equations which are of great interest. One is John's counterexamples $\Box u=u_t^2$ or $\Box u=u_t u_{tt}$, and the other is the 3D compressible Euler equations with no vorticity. We explicitly construct global solutions to the corresponding geometric reduced systems and show that these global solutions satisfy the required pointwise bounds. As a result, there exists a large family of nontrivial global solutions to these two types of equations.