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Geometric analysis of 1+1 dimensional quasilinear wave equations

Leonardo Enrique Abbrescia, Willie Wai Yeung Wong

Abstract

We prove global well-posedness of the initial value problem for a class of variational quasilinear wave equations, in one spatial dimension, with initial data that is not-necessarily small. Key to our argument is a form of quasilinear null condition (a "nilpotent structure") that persists for our class of equations even in the large data setting. This in particular allows us to prove global well-posedness for $C^2$ initial data of moderate decrease, provided the data is sufficiently close to that which generates a simple traveling wave. We take here a geometric approach inspired by works in mathematical relativity and recent works on shock formation for fluid systems. First we recast the equations of motion in terms of a dynamical double-null coordinate system; we show that this formulation semilinearizes our system and decouples the wave variables from the null structure equations. After solving for the wave variables in the double-null coordinate system, we next analyze the null structure equations, using the wave variables as input, to show that the dynamical coordinates are $C^1$ regular and covers the entire space-time.

Geometric analysis of 1+1 dimensional quasilinear wave equations

Abstract

We prove global well-posedness of the initial value problem for a class of variational quasilinear wave equations, in one spatial dimension, with initial data that is not-necessarily small. Key to our argument is a form of quasilinear null condition (a "nilpotent structure") that persists for our class of equations even in the large data setting. This in particular allows us to prove global well-posedness for initial data of moderate decrease, provided the data is sufficiently close to that which generates a simple traveling wave. We take here a geometric approach inspired by works in mathematical relativity and recent works on shock formation for fluid systems. First we recast the equations of motion in terms of a dynamical double-null coordinate system; we show that this formulation semilinearizes our system and decouples the wave variables from the null structure equations. After solving for the wave variables in the double-null coordinate system, we next analyze the null structure equations, using the wave variables as input, to show that the dynamical coordinates are regular and covers the entire space-time.

Paper Structure

This paper contains 13 sections, 6 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. The acoustic metric and its geometry
  3. Simple wave solutions
  4. Perturbed system
  5. Statement of the main theorem
  6. Initial data and gauge fixing
  7. Global existence of the semilinear system
  8. Iteration scheme
  9. Range of $T$
  10. Contraction
  11. Equation for $\xi$
  12. Jacobian bounds
  13. Conclusion

Key Result

Theorem 1.3

The initial value problem for eq:mainwave is globally well-posed, provided the initial data is of moderate decrease and is sufficiently close (in $C^2$ norm) to that of a simple traveling wave.

Theorems & Definitions (15)

  • Theorem 1.3: Rough statement
  • Example 2.8
  • Remark 2.20
  • Remark 2.23
  • Remark 3.7
  • Remark 4.3
  • Theorem 5.2
  • Remark 6.3
  • Lemma 6.6
  • Proposition 7.1
  • ...and 5 more