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 \begin{document}$ 2s $\end{document} with \begin{document}$ s\in [n/4,1) $\end{document} , which seems to be sharp for the validity of the main results of the paper; here \begin{document}$ n = 2,3 $\end{document} is the dimension of space. We prove global well-posedness in \begin{document}$ C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2})) $\end{document} whenever the initial data \begin{document}$ u_0\in D(A) $\end{document} , where \begin{document}$ A $\end{document} 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.