Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus

2021/02/04 by Tristan Roy, Roy, Tristan
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2102.02689

openalex publication_date 2021/02/04 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution in one direction of time that is infinitely smooth and the problem in not well-posed in the other direction. Under other conditions on the nonlinearity and the data, we prove that the equation behaves like a dispersive one: there exists a unique local solution in both directions of time.

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