Regularity and stability of finite energy weak solutions for the Camassa-Holm equations with nonlocal viscosity (Q2225909)

This paper deals with the Camassa-Holm equations with nonlocal viscosity: \[ \begin{array}{lll} &v_t +u\cdot\nabla v+v\cdot \nabla u^T +\nabla p=-\nu(-\Delta)^sv,&t>0,\,x\in\mathbb{R}^n,\\ &u-\alpha^2 \Delta u=v,&t>0,\,x\in\mathbb{R}^n,\\ &\text{div}(u)=0,&t>0,\,x\in\mathbb{R}^n,\\ &v(0,x)=v_0(x),&x\in\mathbb{R}^n. \end{array} \] with \[\frac{n}{4}\le s < 1.\] The authors consider regular initial data and prove that the finite energy weak solutions are regular for all time and are stable with respect to the initial data.