Non-uniform dependence on initial data for the Whitham equation

2016/01/31 by Mathias Nikolai Arnesen, Arnesen, Mathias Nikolai
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons #math.AP

paper · pdf · doi:10.48550/arxiv.1602.00250

19 pages; improved results and presentation

openalex publication_date 2016/01/31 · arxiv created 2016/09/26 · arxiv updated 2016/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem ∂_t u+u∂_x u+L(∂_x u) =0,
u(0,x)=u₀(x) on the torus and on the real line for a class of Fourier multiplier operators L, and prove that the solution map u₀↦ u(t) is not uniformly continuous in H^s(\mathbbT) or H^s(ℝ) for s>(3)/(2). Under certain assumptions, the result also hold for s>0. The class of equations considered includes in particular the Whitham equation and fractional Korteweg-de Vries equations and we show that, in general, the flow map cannot be uniformly continuous if the dispersion of L is weaker than that of the KdV operator. The result is proved by constructing two sequences of solutions converging to the same limit at the initial time, while the distance at a later time is bounded below by a positive constant.

Citations

• The Whitham Equation as a Model for Surface Water Waves
• Nonlocal models for nonlinear, dispersive waves

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