On blow-up shock waves for a nonlinear PDE associated with Euler equations

2009/02/11 by В. А. Галактионов, V. A. Galaktionov, Galaktionov, V. A.
Engineering · Mathematics · #35K55 #35K65 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #math.AP #msc:35K55 #msc:35K65

paper · pdf · doi:10.48550/arxiv.0902.1840

16 pages, 8 figures

arxiv created 2009/02/11 · openalex publication_date 2009/02/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A second-order PDE is derived from Euler's equaitons under certain assumptions. It is shown that this PDE admits shock and rarefaction waves, and that a single point gradient blow-up admits a unique similarity extension after blow-up that settles uniqueness/entropy issues for such equations.

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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