On the wave equation with quadratic nonlinearities in three space dimensions

2009/12/22 by Axel Gruenrock, Gruenrock, Axel
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations #math.AP #msc:35L70

paper · pdf · doi:10.48550/arxiv.0912.4400

arxiv created 2009/12/22 · arxiv updated 2010/01/14

Abstract

The Cauchy problem for the nonlinear wave equation \Box u=(∂ u)², u(0)=u₀, u_t(0)=u₁ in three space dimensions is considered. The data (u₀,u₁) are assumed to belong to \widehatH^r_s(\R³) × \widehatH^r_s-1(\R³), where \widehatH^r_s is defined by the norm \nf\widehatH^r_s := \n< ξ> ^s\widehatfL^r'_ξ, < ξ>=(1+|ξ|²)^\frac12, (1)/(r)+(1)/(r')=1. Local well-posedness is shown in the parameter range 2 ≥ r >1, s > 1 + (2)/(r). For r=2 this coincides with the result of Ponce and Sideris, which is optimal on the H^s-scale by Lindblad's counterexamples, but nonetheless leaves a gap of \frac12 derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for \Box u = ∂ u² are obtained, too.

Citations

• On the regularity properties of a model problem related to wave maps

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