A Large Data Regime for non-linear Wave Equations

2012/10/07 by Jinhua Wang, Wang, Jinhua, Pin Yu +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Nonlinear Waves and Solitons #math.AP

paper · pdf · doi:10.48550/arxiv.1210.2056

44 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1207.5591

arxiv created 2012/10/07 · openalex publication_date 2012/10/07 · arxiv updated 2012/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For semi-linear wave equations with null form non-linearities on ℝ³⁺¹, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future. We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an specific incoming null geodesic in such a way that almost all of the energy is confined in a tubular neighborhood of the geodesic and almost no energy radiating out of this tubular neighborhood.

Citations

• Uniform decay estimates and the lorentz invariance of the classical wave equation
• Long time solutions for wave maps with large data

Cited by

• Global existence of smooth solution to evolutionary Faddeev model with short-pulse data
• Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms
• Global solutions of nonlinear wave equations with large energy

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