We consider the n-dimensional (\(n=2,3\)) Camassa–Holm equations with nonlocal diffusion of type \((-\,\Delta )^{s}, \ \frac{n}{4}\le s<1\). 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 \(s=\frac{n}{4}\) contains extra difficulties and a smallness assumption on the initial data is imposed.

我们考虑具有\（（-\，\ Delta）^ {s}，\ \ frac {n} {4} \类型的非局部扩散的n维（\（n = 2,3 \））Camassa–Holm方程le s <1 \）。在甘等人。（离散Contin Dyn Syst 40（6）：3427-3450，2020年），建立了有限能量弱解的全局时间存在性和唯一性。在本文中，我们证明了使用规则的初始数据，有限能量弱解的确在所有时间都是规则的。而且，弱解相对于初始数据是稳定的。主要困难在于在存在分数拉普拉斯扩散的情况下建立更高阶的均匀估计。为此，我们需要探索分数分数的Sobolev型不等式和分数导数的双线性估计。关键情况\（s = \ frac {n} {4} \）包含额外的困难，并且对初始数据的假设很小。