Global smooth solutions to 4D quasilinear wave equations with short pulse initial data

TL;DR

This paper proves the global existence of smooth solutions for 4D quasilinear wave equations with short pulse initial data satisfying the first null condition, extending understanding of wave behavior under specific initial conditions.

Contribution

It establishes the first null condition as necessary and sufficient for global smooth solutions with short pulse data in 4D quasilinear wave equations.

Findings

Global smooth solutions exist for equations satisfying the first null condition.

Solutions to certain physical models like Chaplygin gases are globally smooth under short pulse data.

Solutions to polytropic gases generally blow up in finite time without the null condition.

Abstract

In this paper, we establish the global existence of smooth solutions to general 4D quasilinear wave equations satisfying the first null condition with the short pulse initial data. Although the global existence of small data solutions to 4D quasilinear wave equations holds true without any requirement of null conditions, yet for short pulse data, in general, it is sufficient and necessary to require the fulfillment of the first null condition to have global smooth solutions. It is noted that short pulse data are extensions of a class of spherically symmetric data, for which the smallness restrictions are imposed on angular directions and along the outgoing directional derivative $\partial_t+\partial_r$, but the largeness is kept for the incoming directional derivative $\partial_t-\partial_r$. We expect that here methods can be applied to study the global smooth solution or blowup problem with short pulse initial data for the general 2D and 3D quasilinear wave equations when the corresponding null conditions hold or not. On the other hand, as some direct applications of our main results, one can show that for the short pulse initial data, the smooth solutions to the 4D irrotational compressible Euler equations for Chaplygin gases, 4D nonlinear membrane equations and 4D relativistic membrane equations exist globally since their nonlinearities satisfy the first null condition; while the smooth solutions to the 4D irrotational compressible Euler equations for polytropic gases generally blow up in finite time since the corresponding first null condition does not hold.