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Uniform estimates for 2D quasilinear wave

Dong Li

Abstract

We consider two-dimensional quasilinear wave equations with standard null-type quadratic nonlinearities. In 2001 Alinhac proved that such systems possess global in time solutions for compactly supported initial data with sufficiently small Sobolev norm. The highest norm of the constructed solution grows polynomially in time. In this work we develop a new strategy and prove uniform boundedness of the highest order norm of the solution for all time.

Uniform estimates for 2D quasilinear wave

Abstract

We consider two-dimensional quasilinear wave equations with standard null-type quadratic nonlinearities. In 2001 Alinhac proved that such systems possess global in time solutions for compactly supported initial data with sufficiently small Sobolev norm. The highest norm of the constructed solution grows polynomially in time. In this work we develop a new strategy and prove uniform boundedness of the highest order norm of the solution for all time.

Paper Structure

This paper contains 8 sections, 11 theorems, 218 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main']}
  4. The case $\alpha_1<\alpha$ and $\alpha_2<\alpha$
  5. The case $\alpha_{2}=\alpha$.
  6. Estimate of the main piece $\alpha_{1}=\alpha$, $\alpha_{2}=0$
  7. further estimates
  8. Estimate of $\| \langle \nabla \rangle^{-1} \partial \Gamma^{\le m+1} u \|_2$

Key Result

Theorem 1.1

Consider eq:we2d with $g^{kij}$ satisfying the standard null condition null1. Let $m\ge 5$ and assume $f_1 \in H^{m+1}(\mathbb R^2)$, $f_2\in H^{m} (\mathbb R^2)$ are compactly supported in the disk $\{|x| \le 1\}$. There exists $\varepsilon_{0}>0$ depending on $g^{kij}$ and $m$ such that if $\|f_1\ Here $\Gamma=\{ \partial_t, \partial_{x_1}, \partial_{x_2}, \partial_{\theta}, t\partial_t +r\parti

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1: The nonlocal norm is well-defined
  • proof
  • Lemma 2.2: Sobolev decay
  • proof
  • Lemma 2.3: Refined Hardy's inequality
  • proof
  • Lemma 2.4
  • proof
  • ...and 12 more