Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to λ-family equations

2021/11/01 by Geng Chen, Chen, Geng, Yannan Shen +3
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.2111.01030

Abstract

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as λ-family equations, where λ is the power of nonlinear wave speed. The λ-family equations include Camassa-Holm equation (λ=1) and Novikov equation (λ=2) modelling water waves, where solutions generically form finite time cusp singularities, or in another word, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent 1- (1)/(2λ). The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.

Citations

• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Wave breaking for nonlinear nonlocal shallow water equations

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• Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations
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