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 ∈ [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.