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

Abstract: In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result. 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. We then apply such a conditional result to Fritz John's counterexample \Box u=u_tu_{tt} and the 3D compressible Euler equations without 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 each of these two types of equations.