On the Viscous Camassa-Holm Equations with Fractional Diffusion

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 $s\in [n/4,1)$, 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 $C_{[0,+\infty )}(D(A))\cap L_{[0,+\infty ),loc}^2(D(A^{1+s/2}))$ whenever the initial data $u_0\in D(A)$, 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.

Local and global well-posedness results have been improved in this version

References in corpus (1)

• Local and global strong solutions for SQG in bounded domains

Cited by in corpus (2)

• Regularity of Solutions of the Camassa-Holm Equations with Fractional Laplacian Viscosity
• Large Time Behavior and Convergence for the Camassa-Holm Equations with Fractional Laplacian Viscosity