On The Well-posedness Of Two Kinds Of Nonlinear Partial Differential Equations

In this paper, we mainly study the well-posedness of two kinds of nonlinear partial differential equations with physic background. The first one is the high-order Camassa–Holm equation and the second one is the Fokker-Planck-Boltzmann equation. The high-order Camassa–Holm equation can be derived as the Euler-Poincar′e differential equation on the Bott-Virasoro group with respect to the H2 metric. The Fokker-Planck-Boltzmann(FPB)equation describes the motion of particles in a thermal bath.This thesis is divided into three chapter. In the first Chapter, we introduce the high-order Camassa–Holm equation and the FPB equation and we also state some preliminaries. In the second Chapter, we consider the Cauchy problem for the high-order Camassa–Holm equation yt+ 2uxy + uyx= 0, y =(1- ?2x)2u, t > 0, x ∈R,u(0, x) = u0(x) ∈ Bs p,r.We establish the local well-posedness for the above problem in Besov spaces Bs p,rwith s > max{7/2, 3 + 1/p}, 1 ≤ p, r ≤ ∞ or s = 7/2, p = 2, r = 1. Then we show that if a weaker Bq p,r-topology is used, then the solution map becomes H¨older continuous. Finally,by construction, we show that for the periodic boundary value problem of the high-order Camassa–Holm equation, the solution map is not uniformly continuous. In Chapter 3,we study the Cauchy problem for the Fokker-Planck-Boltzmann equation?tf + ξ · xf = Q(f, f) + ξ·(ξf) + ?ξf, t > 0, ξ ∈R3ξ, x ∈R3x,f(0, ξ, x) = f0(ξ, x), ξ ∈R3ξ, x ∈R3x.We prove that when the initial data is a small perturbation of the equilibrium state, under the Grad’s angular cuto assumption and for the hard potential case, the above problem has a unique global solution in L2ξ(Bs2,r) where s > 3/2, 1 ≤ r ≤ 2 or s = 3/2, r = 1.