On the viscous Camassa-Holm equations with fractional diffusion

Zaihui Gan, Fanghua Lin, Jiajun Tong

Research output: Contribution to journalArticle

Abstract

We study Cauchy problem of a class of viscous Camassa-Holm equations (or Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and in the whole space in two and three dimensions. Order of the fractional diffusion is assumed to be 2s with s ∈ [n/4, 1), which seems to be sharp for the validity of the main results of the paper; here n = 2, 3 is the dimension of space. We prove global well-posedness in C[0,+)(D(A)) ∩ L2 [0,+∞),loc(D(A1+s/2)) whenever the initial data u0 ∈ D(A), where A is the Stokes operator. We also prove that such global solutions gain regularity instantaneously after the initial time. A bound on a higher-order spatial norm is also obtained.

Original languageEnglish (US)
Pages (from-to)3427-3450
Number of pages24
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume40
Issue number6
DOIs
StatePublished - 2020

Keywords

  • Fractional diffusion
  • Global well-posedness
  • Improved regularity
  • Lagrangian averaged Navier-Stokes equations
  • Viscous Camassa-Holm equations

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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