On the global behavior of weak null quasilinear wave equations

2018/04/13 by Yu Deng, Fabio Pusateri, Deng, Yu +1 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1804.05107

48 pages

arxiv created 2018/04/13 · arxiv updated 2018/04/17

Abstract

We consider a class of quasilinear wave equations in 3+1 space-time dimensions that satisfy the "weak null condition" as defined by Lindblad and Rodnianski \citeLR1, and study the large time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is -∂_t² u + (1+u) Δu = 0. Global solutions for such equations have been constructed by Lindblad \citeLindblad1,Lindblad2 and Alinhac \citeAlinhac1. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solutions close to the light cone, including the blow-up at infinity.

Citations

• Nonlinear fractional Schrödinger equations in one dimension
• The null condition for quasilinear wave equations in two space dimensions I
• Uniform decay estimates and the lorentz invariance of the classical wave equation

Cited by

• Late-time tail for a scalar quasilinear wave equation satisfying the weak null condition
• Space-time resonances in the spatiotemporal spectrum of nonlinear dispersive waves

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