Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy

2009/02/10 by Victor A. Galaktionov, V. A. Galaktionov, Galaktionov, V. A.
Mathematics · Physics and Astronomy · #35K40 #35K55 #35K65 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #math.AP #msc:35K40 #msc:35K55 #msc:35K65

paper · pdf · doi:10.48550/arxiv.0902.1635

19 pages, 8 figures

arxiv created 2009/02/10 · openalex publication_date 2009/02/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that third-order 1D nonlinear dispersion equations admit single point gradient catastrophe, described by blow-up-type similarity solutions. After blow-up, the solutions admit shock wave-type self-similar extensions. Snce such extensions are not unique, this implies the principle nonuniqueness of shock-type solutions and also nonexistence of any entropy-type description of proper unique solutions. A difficult free-boundary setting, with extra conditions specified on shocks, are necessary to restore uniqueness in such problems.

Citations

• Third-order nonlinear dispersion PDEs: shocks, rarefaction, and blow-up waves
• Nonlinear dispersion equations: smooth deformations, compactons, and extensions to higher orders

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