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

Bingbing Ding, Zhouping Xin, Huicheng Yin

TL;DR

The paper proves the global existence of smooth solutions to 4D quasilinear wave equations with short-pulse initial data under the first null condition. It develops a geometric-energy framework based on an optical function $u$ and the inverse foliation density $\mu$ to control outgoing characteristics and prevent shock formation, while constructing a global Goursat problem to cover the interior region. The analysis combines bootstrap arguments, refined $L^\infty$ and $L^2$ estimates, transport and elliptic equations on null foliations, and careful treatment of error terms to yield decay rates $|\phi| \le C\delta^{(4-5\varepsilon_0)/3} t^{-1/2}$ and $|\partial\phi| \le C\delta^{1-\varepsilon_0} t^{-3/2}$ for all $t\ge 1$. The results provide a robust route to global regularity for large-short-pulse data in high dimensions and illuminate the roles of null conditions in preventing blow-up in 4D quasilinear wave dynamics, with direct applications to Chaplygin gas Euler flows and membrane equations.

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.

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

TL;DR

The paper proves the global existence of smooth solutions to 4D quasilinear wave equations with short-pulse initial data under the first null condition. It develops a geometric-energy framework based on an optical function and the inverse foliation density to control outgoing characteristics and prevent shock formation, while constructing a global Goursat problem to cover the interior region. The analysis combines bootstrap arguments, refined and estimates, transport and elliptic equations on null foliations, and careful treatment of error terms to yield decay rates and for all . The results provide a robust route to global regularity for large-short-pulse data in high dimensions and illuminate the roles of null conditions in preventing blow-up in 4D quasilinear wave dynamics, with direct applications to Chaplygin gas Euler flows and membrane equations.

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 , but the largeness is kept for the incoming directional derivative . 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.
Paper Structure (25 sections, 39 theorems, 399 equations)

This paper contains 25 sections, 39 theorems, 399 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem and main results
  3. Sketch for the proof of Theorem \ref{['main']}
  4. Local existence of the solution $\phi$ for $t\in [1, 1+2\delta]$
  5. Global existence of the solution $\phi$ in $A_{2\delta}$
  6. Global existence of solution $\phi$ inside $B_{2\delta}$
  7. Organization of the paper
  8. Notations
  9. On the short pulse initial data
  10. Local existence
  11. Some preliminaries
  12. The related geometry and definitions
  13. Basic equalities in the null frames
  14. Bootstrap assumptions on $(\phi,\partial\phi)$ near $C_0$ and some related estimates
  15. $L^\infty$ estimates on the higher order derivatives of $(\phi,\partial\phi)$ and some other quantities near $C_0$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Consider the Cauchy problem for the 4D quasilinear wave equation quasi with the short pulse initial data Y1-1 satisfying Y-00. Assume that g00 holds and the first null condition null is satisfied. Then for $\delta>0$ suitably small, then there exists a global smooth solution $\phi$ to the problem qu for all time $t\ge 1$, where $C>0$ is a uniform constant independent of $\delta$ and $\varepsilon_0

Theorems & Definitions (87)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • ...and 77 more