Regularity and stability of finite energy weak solutions for the Camassa-Holm equations with nonlocal viscosity

We consider the n-dimensional (n = 2, 3) Camassa-Holm equations with nonlocal diffusion of type (-Delta)(s), n/4 <= s < 1. In Gan et al. (Discrete Contin Dyn Syst 40(6):3427-3450, 2020), the global-in-time existence and uniqueness of finite energy weak solutions is established. In this paper, we show that with regular initial data, the finite energy weak solutions are indeed regular for all time. Moreover, the weak solutions are stable with respect to the initial data. The main difficulty lies in establishing higher order uniform estimates with the presence of the fractional Laplacian diffusion. To achieve this, we need to explore suitable fractional Sobolev type inequalities and bilinear estimates for fractional derivatives. The critical case s = n/4 contains extra difficulties and a smallness assumption on the initial data is imposed.