Regularity of Solutions of the Camassa-Holm Equations with Fractional Laplacian Viscosity

Abstract

We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $\displaystyle s=\frac{n}{4}$ with $n=2,3$, we will make extra efforts for acquiring the expected estimates as obtained in the case $\displaystyle \frac{n}{4}<s<1$. By the aid of the fractional Leibniz rule and the nonlocal version of Ladyzhenskaya's inequality, we prove the existence, uniqueness and regularity to the Camassa-Holm equations under study by the energy method and a bootstrap argument, which rely crucially on the fractional Laplacian viscosity. In particular, under the critical case $s=\dfrac{n}{4}$, the nonlocal version of Ladyzhenskaya's inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the desired results concernig existence, uniqueness and regularity.