On the Viscous Camassa-Holm Equations with Fractional Diffusion

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 $C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2}))$ 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.