Oleinik type estimates for the Ostrovsky-Hunter eequation

2014/03/10 by Giuseppe Maria Coclite, G. M. Coclite, Coclite, G. M. +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #advanced mathematical theories #math.AP

paper · pdf · doi:10.48550/arxiv.1403.2349

arxiv created 2014/03/10 · arxiv updated 2016/10/05

Abstract

The Ostrovsky-Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. In this paper we study the well-posedness for the Cauchy problem associated to this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities by an Oleinik-type estimate and prove uniqueness via a nonlocal adjoint problem. An implication is that a shock wave in an entropy weak solution to the Ostrovsky-Hunter equation is admissible only if it jumps down in value (like the inviscid Burgers equation).

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