Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation

2015/02/09 by Alberto Bressan, Geng Chen, Bressan, Alberto +1 · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1502.02611

openalex publication_date 2015/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The paper is concerned with conservative solutions to the nonlinear wave equation u_tt - c(u)(c(u) u_x)_x = 0. For an open dense set of C³ initial data, we prove that the solution is piecewise smooth in the t-x plane, while the gradient u_x can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.

Citations

• Structurally Stable Singularities for a Nonlinear Wave Equation

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