Well-posedness And Asymptotic Behavior For Several Kinds Of Evolution Equations

In this dissertation,we mainly study well-possedness and asymptotic behavior of some kinds of evolution equations:the nonlinear magnetohydrodynamic(MHD)and linear KdV-BO,Boussinesq and Ostrovsky equations.In the second chapter,we prove the well-posedness of 3D MHD system with a damping term.By using Diophantine condition,we get a high-order Poincare inequality in a specific direction and obtain the uniform estimate of the solutions.Then we prove the global existence of the solutions of the small initial value promblem by using the continuity argument.At the same time,we get the decay of the solutions.In the third chapter,we mainly study the Carleson’s problem of the KdV-BO equation and obtain the almost surly everywhere pointwise convergence of the equation with randomized initial data.In the fourth chapter,we present the Hausdorff dimension of the divergence set for the Ostrovsky equation.In the five chapter,we study pointwise convergence problem of Boussinesq equation with random data.In Chapter 2,we consider the Cauchy problem of MHD equation in T3:By using Diophantine condition,we get a high-order Poincaré inequality in a specific direction and prove for any γ≥ 4r+7 with r>2,let(u0,b0)∈ Hγ(T3)and satisfying∫T3 u0dx=∫T3 b0 dx=0.If there exists a small constant ε such that‖u0‖Hγ+‖b0‖Hγ≤ε.Then the system(0-3)admits a global solution(u,b)∈ C([0,∞);Hγ).Moreover,when r+4 ≤α<γ,the solutions satisfy the following decay property‖u(t)‖Hα+‖b(t)‖Hα≤C(1+t)-3(γ-α)/2(γ-r-4).In Chapter 3,we investigate the pointwise convergence problem for the Korteweg-de Vries-Benjamin-Ono equation where γβ>0.We prove that the solution u(x,t)=Utf(x)converges pointwisely to the initial data f(x)for a.e.x ∈ R when f∈ Hs(R)with ≤ 1/4,and that the Hausdorff dimension of the divergence set of points of the solution isα1,U(s)=1-2s,1/4≤s≤1/2.We also obtain the stochastic continuity for the initial data with much less regularity,i.e.for a large class of the initial data in L2(R),via the randomization technique,we get uω(t,x)=Utfω(x)→fω(x),a.s.ω∈Ω.In Chapter 4,we mainly study the refined Carleson’s problem of the free Ostrovsky equation where(x,t)∈ R × R and f∈Hs(R),and present the Hausdorff dimension of the divergence set for the Ostrovsky equation(0-5)α1,U(s)=1-2s,1/4≤s≤1/2.In Chapter 5,we consider the pointwise convergence problem of the free induced Boussinesq equation with random data in L2(Rn)and L2(Tn).We get the results similar to those in Chapter 3,i.e.uω(t,x)→fω(x),a.s.ω∈Ω.