Large time behavior and convergence for the Camassa–Holm equations with fractional Laplacian viscosity

Abstract

In this paper, we consider the n-dimensional ( $$n=2,3$$ ) Camassa–Holm equations with fractional Laplacian viscosity in the whole space. In contrast to the Camassa–Holm equations without any nonlocal effect, much less has been known on the large time behavior and convergences of solutions. Here we study first the large time behavior of solutions, then consider the relation between the equations under consideration and the imcompressible Navier–Stokes equations with fractional Laplacian viscosity (INSF). By applying the fractional Leibniz chain rule and the fractional Gagliardo–Nirenberg–Sobolev type estimates, the high and low frequency splitting method and the Fourier splitting method, we shall establish the large time non-uniform decays and algebraic rate decays of solutions. In the critical case $$s=\dfrac{n}{4}$$ , the nonlocal version of Ladyzhenskaya’s inequality along with the smallness of initial data in suitable Sobolev spaces is needed. In addition, by estimates for the fractional heat kernels, we prove that the solutions to the Camassa–Holm equations with nonlocal viscosity converge strongly as the filter parameter $$\alpha \rightarrow ~0$$ to solutions of the equations INSF.