Regularity of weak solutions for the fractional Camassa-Holm equations

This paper focuses on the n-dimensional (n = 2, 3) Camassa-Holm equations with non-local diffusion of type (-Delta)(s). In Gan et al. (Calc Var Partial Differ Equ. 60, 2021), they proved that with regular initial data, the finite energy weak solutions are indeed regular for all time if n/4 < s < 1. The purpose of this paper is to improve the result of Gan et al. Actually, we will establish the following regularity criterion: If del u is an element of L-q ([0, T]; L-r (R-n)) with n/r +2s/q <= 2s, r > n/2s, then v is regular. As a corollary, we can obtain that if n-2/4 < s < 1, then the finite energy weak solutions of the fractional Camassa-Holm equations are regular. In addition, by partially appealing to some ideas used in partial regularity theory, when n = 3, we can prove that if 1/2< s < 1, then the suitable weak solution of the fractional Camassa-Holm equations must be bounded.