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 (- Δ) 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 ∇ u ∈ L q ([ 0 , T ] ; L r (R n)) <graphic href="33_2022_1793_Article_Equ39.gif"></graphic> with n r + 2 s q ≤ 2 s , r > n 2 s , <graphic href="33_2022_1793_Article_Equ40.gif"></graphic> 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. [ABSTRACT FROM AUTHOR]