Global Well-posedness For Two Kinds Of Systems In Incompressible Fluid Mechanics

In this paper, we mainly consider two kinds of important systems in fluid me-chanics:the nonhomogeneous incompressible Navier-Stokes equations and the nonho-mogeneous incompressible MHD equations. The Navier-Stokes equations are a kind of equations of motion which describe the momentum conservation of the viscous in-compressible fluid. The Magnetohydrodynamics (MHD) equations mainly describe the interaction between the magnetic field and electrically conducting fluids (plasmas, liq-uid metals, etc.).The article is divided into three parts. In the first part, we mainly consider the global well-posedness of the Navier-Stokes equations in the critical Besov spaces, when the initial data satisfy a kind of "nonlinear smallness" condition. Also, we construct a class of initial data which satisfy that "nonlinear smallness" condition, while, the norms of the initial data (even all components of them) can be arbitrarily large in the critical Besov spaces. In the second part, we study the global well-posedness of the nonhomogeneous incompressible MHD equations in the critical Besov spaces. Similarly, we construct a class of initial data which satisfy some "nonlinear smallness" condition while the components of which can also be arbitrarily large in the critical Besov spaces. In the third part, we mainly study the global existence, uniqueness and the large-time behavior of the strong solutions of the variable viscosity and variable conductivity MHD equations in both vacuum and non-vacuum cases when the initial data are suitably small in some sense.The main results are as follows.In Chapter Two, we study the Cauchy problem of the following nonhomogeneous incompressible Navier-Stokes equations: Firstly, when the density ρ> 0 (set a:= 1/ρ - 1), we prove that when the initial data (a0,μ0) in the critical Besov spaces Bn/qq,1(Rn)×Bn/p-1p,1(Rn) ((p,q) ∈ [1,2n) [1, ∞),-1/n≤ 1/p-1/q≤1/n) satisfy the following "nonlinear smallness" condition the equations (0-5) are global well-posed. Secondly, to illustrate the rationality of the "nonlinear smallness" condition (0-6), we construct a class of initial data (εα0,u0,ε) which satisfy (0-6). While, the norms of all the components of the velocity u0,ε can be arbitrarily large in Bn/pp,1 (Rn) (n<p<2n). Specifically, we take out a free heat equation to undertake the initial data of the velocity. Then, for the residual of the Navier-Stokes equations, we take full advantage of the weighted Chemin-Lerner norms and the estimates of commutator to get the estimates on the transport equation. Then we get the estimates of the Stokes equation, which lead to the estimates of the velocity field, the magnetic field and the pressure. At last, according to the continuous method, by choosing a proper weighted function, we can close the energy estimates.In Chapter Three, we study the Cauchy problem of the following 3D nonhomoge-neous incompressible MHD equations: Firstly, we show that there exists a unique global solution to the equations (0-7) provided that the initial data (a0,u0,b0) (a0:=1/ρ0-1) in the critical Besov spaces B3/qq,1(R3)×B-1+3/pp,1(R3)× B-1+3/pp,1(R3) satisfy a "nonlinear smallness" condition where for all 1<q≤p<6,1/q-1/p<1/3. Secondly, similar to the situation of the Navier-Stokes equations, we construct a class of initial data (εao, u0,ε,b0,ε), which satisfy (0-8), but the norms of uO,ε,bO,ε (even all their components) can be arbitrarily large in B-1p,1+3/p(R3).In Chapter Four, we consider the initial boundary value problem of the incom-pressible MHD equations with variable viscosity and variable conductivity in a smooth bounded domain Ω (?)R3. When the density, the velocity field and the magnetic field satisfy the following initial and boundary conditionswe establish the global existence and uniqueness of strong solutions in both vacuum and non-vacuum cases when the initial velocity field and magnetic field are suitably small in some sense without any smallness condition on the initial density. In addition, we also get some results of the large-time behavior of the solutions in both two cases. More precisely, we propose the following a priori hypothesis on the viscous coefficient and magnetic field dissipation coefficient:According to the energy method, we get the time-weighted global a priori estimates which are independent of the condition ρ≥ρ≥ O. And then, the global a priori estimate of｜｜(▽μ,▽H)｜｜L1((o,T),L∞) is obtained, which is the key ingredient to keeping the propagation of the regularities of Vp and ▽ρ and▽μ(ρ). At last, once the a priori hy-pothesis is proved, we can close the global energy estimate by the continuous method respectively in both two cases. At the same time, from the time-weighted global a priori estimates, we get the large-time behavior of the solutions in both two cases.