The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded

TL;DR

This paper proves that the top-order energy of solutions to certain two-dimensional quasilinear wave equations remains uniformly bounded over time, contradicting previous conjectures of unbounded growth.

Contribution

It establishes the global boundedness of the top-order energy for quasilinear wave equations in two dimensions, challenging prior beliefs of inevitable energy growth.

Findings

Top-order energy remains uniformly bounded in time.

Contradicts Alinhac's blowup-at-infinity conjecture.

Provides new insights into energy behavior of 2D wave equations.

Abstract

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions, provided that the initial data are compactly supported and sufficiently small in Sobolev norm. In this work, Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions. A natural question then arises whether the time-growth is a true phenomena, despite the possible conservation of basic energy. Analogous problems are also of central importance for Schr\"odinger equations and the incompressible Euler equations in two space dimensions, as studied by Bourgain, Colliander-Keel-Staffilani-Takaoka-Tao, Kiselev-Sverak, and others. In the present paper, we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time, which is opposite to Alinhac's blowup-at-infinity conjecture.