Regularity and lifespan of small solutions to systems of quasi-linear wave equations with multiple speeds,I: almost global existence

TL;DR

This paper proves almost global existence of small solutions for symmetric systems of wave equations with multiple speeds in 2D and 3D, overcoming key analytical challenges with novel weighted inequalities.

Contribution

It introduces new weighted inequalities as substitutes for classical Klainerman-Sobolev inequalities to establish a priori bounds for low-regularity solutions.

Findings

Almost global existence for small initial data in 2D and 3D

Development of weighted inequalities as analytical tools

Overcoming the absence of $H^1$-$L^p$ Klainerman-Sobolev inequality

Abstract

In this paper, we show almost global existence of small solutions to the Cauchy problem for symmetric system of wave equations with quadratic (in 3D) or cubic (in 2D) nonlinear terms and multiple propagation speeds. To measure the size of initial data, we employ a weighted Sobolev norm whose regularity index is the smallest among all the admissible Sobolev norms of integer order. We must overcome the difficulty caused by the absence of the $H^1$-$L^p$ Klainerman-Sobolev type inequality, in order to obtain a required a priori bound in the low-order Sobolev norm. The introduction of good substitutes for this inequality is therefore at the core of this paper. Using the idea of showing the well-known Lady\v{z}enskaja inequality, we prove some weighted inequalities, which, together with the generalized Strauss inequality, play a role as the good substitute.