Global existence for 2-D wave maps equation in exterior domains

2025/01/29 by Hou, Fei, Yin, Huicheng, Yuan, Meng
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.18646

Abstract

In the paper [H. Kubo, Global existence for exterior problems of semilinear wave equations with the null condition in 2D, Evol. Equ. Control Theory 2 (2013), no. 2, 319-335], for the 2-D semilinear wave equation system (∂_t²-Δ)v^I=Q^I(∂_tv, ∇_xv) (1≤ I≤ M) in the exterior domain with Dirichlet boundary condition, it is shown that the small data smooth solution v=(v¹, ⋅⋅⋅, v^M) exists globally when the cubic nonlinearities Q^I(∂_tv, ∇_xv)=O(|∂_tv|³+|∇_xv|³) satisfy the null condition. We now focus on the global Dirichelt boundary value problem of 2-D wave maps equation with the form \Box u^I=∑_J,K,L=1^MC_IJKLu^JQ₀(u^K,u^L) (1≤ I≤ M) and Q₀(f,g)=∂_tf∂_tg-∑_j=1²∂_jf∂_jg in exterior domain. By establishing some crucial classes of pointwise spacetime decay estimates for the small data solution u=(u¹, ⋅⋅⋅, u^M) and its derivatives, the global existence of u is shown.

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