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

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.

19 pages, 8 figures

References in corpus (3)

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

Cited by in corpus (4)

• Third-order nonlinear dispersion PDEs: shocks, rarefaction, and blow-up waves
• Nonlinear dispersion equations: smooth deformations, compactons, and extensions to higher orders
• Eigenfunctions and Very Singular Similarity Solutions of Odd-Order Nonlinear Dispersion PDEs
• On blow-up shock waves for a nonlinear PDE associated with Euler equations