Large Time Behavior And Convergence Of Solutions For The Camassa-Holm Equations With Fractional Diffusion Viscosity

In the recent years,with the establishing of fractional Leibniz chain rules and some other estimates on fractional-order derivatives,many researchers pay more attention to the study of the nonlinear partial differential equations with Nonlocal operator viscosity and obtain many important and interesting research results.However,there are few results on the study of the Camassa-Holm equations with fractional diffusion viscosity(FCCH).In this paper,We study the large time behavior and convergence of solutions to the n-dimensional(n=2,3)Camassa-Holm equations with fractional diffusion viscosity in the whole space:In Chapter 1,we introduce some applied science background and some related previous important progress for the viscous Camassa-Holm equations.Then we state our main results concerning the large time behaviour and convergence of solutions to the Cauchy problem for the Camassa-Holm equations under consideration in this thesis.In Chapter 2,we give some preliminary ingredients which are key to the study of the main results of this thesis.In Chapter 3 and Chapter 4,we study the Cauchy problem for the equations(FCCH).By applying the fractional Leibniz chain rule,the fractional Gagliardo-NirenbergSobolev type estimates,the high-low frequency splitting method,and the Fourier splitting method,we establish the large time behavior(non-uniform decay and algebraic decay)of solutions to the Cauchy problem of the equations(FCCH)in the whole space.In Chapter 3,by splitting the energy into low and high frequency parts,using a cut-off function and the generalized energy inequalities,we show that the non-uniform decay estimates of solutions to the Cauchy problem of the Camassa-Holm equations with fractional viscosity(FCCH)as t→+∞.In Chapter 4,with the Fourier splitting method and the bootstrap argument,we establish the algebraic decay of solutions to the Cauchy problem of the equations(FCCH).In Chapter 5,we discuss the relationship of solutions between the Camassa-Holm equations with fractional diffusion viscosity and the incompressible Navier-Stokes equation with fractional Laplace viscosity.By means of the fractional heat kernel estimates,Leray projection and the related estimates of linear nonlocal operator (?)t+(-Δ)γ0/2,we show that if the solutions of the incompressible Navier-Stokes equations with fractional Laplace viscosity:is known to be sufficiently regular,then the solution of the Camassa-Holm equations with fractional diffusion viscosity(FCCH)converges strongly to the one of the equations(NS)in L∞([0,T],Lq(Rn)as α→0.