Mathematical Methods For Incompressible Hydrodynamic And Magneto-hydrodynamic Equations

In this paper, the incompressible hydrodynamics and magneto-hydrodynamics equations are studied from the mathematical view point. L. Euler formulated the Euler equation to describe the motion of an ideal fluid (inviscid fluid) in 1755 [26] Navier derived, in 1822, the Navier-Stokes equationto describe the motion law of viscous fluid [76]. Where v is the viscous coefficient, and Stokes made clear the physical meaning of the viscous coefficient v in 1845 [89]. During more than 200 years, the research of Euler and Navier-Stokes equations has been developed rapidly, and extensively applied to engineering, in particular, shipping industry, aeronautics and astronautics, and meteorology. Many researchers studied extensively Euler and Navier-Stokes equations from different view points. The new studying methods are continuously put forward, and the mathematical theories get continuously abundant. For example, in 1933, Leray used the energy estimates to prove the existence of weak solutions to Navier-Stokes for the first time [58]. Subsequently, many papers were devoted to the research of regular theories of Leray-Hopf weak solutions [84, 85, 5, 6, 65, 54, 3, 22, 52, 53, 56]. Kato and Fujita gave an approach based on semi-group to construct the strong solutions on a strong space directly in 1962 [45], and Kato established the well-posedness for Navier-Stokes equations on space L~n(R~n) by the semi-group method in 1984 [42]. Kato's method has been appliedto different functional spaces extensively in recent years [45, 42, 60, 38, 11, 12, 13, 28, 44, 91, 2, 97, 70, 18, 50].In this paper the hydrodynamic and magneto-hydrodynamic equations are studied from two different view points. On the one hand, we discuss the well-posedness of the Cauchy problems on different functional spaces; On the other hand, the blow-up criteria of smooth solutions to the magneto-hydrodynamic equations are studied. This thesis is divided into four parts.Part one. Studying the global well-posedness of solutions to the Navier-Stokes equations on weak Morrey spaces for small initial data and the existence of self-similar solutions.We construct the weak Morrey spaces, which generalize Morrey and Lorentz spaces and have many advantages. On the one hand, they include Morrey spaces M|(R"), I < p < q < oo and all Lorentz spaces Lp](?(Rn), 1 < p < q < oo. On the other hand, weak Morrey space may admit the functions that satisfy self-similar structure. In order to establish the well-posedness of solutions to the Navier-Stokes equations on weak Morrey spaces, We define the weak Morrey spaces M*O0 = L*(Rn) (In particular, At*0(Rn) = LPiOO for p > 1), and some fundamental properties of them are studied; Subsequently, we prove that heat operator U(t) = etA and Calderon-Zygmund singular integral operator are bounded linear operators on weak Morrey spaces, and establish the bilinear estimate of the nonlinear term of the integral equations on weak Morrey spaces. Finally, by means of Kato's method and Banach contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces M*A(M?)(1 < p < n) is global well-posed, provided that the initial data are appropriately small. Moreover, we also obtain the existence and uniqueness of self-similar solution, because the weak Morrey space M*n_p(Rn) can admit the singular initial data with self-similar structure.Part two. Consider the incompressible magneto-hydrodynamic systemwith initial dataf it(ar,O) = uo(x),{ (0-10)[b(x,O) = bo(x),where x eW1, t>0, and n > 2 is the space dimension. Hereu(x,t) = (u1{x,t),u2{x,t),---,un{x,t)) is the velocity of the fluid flows, andb(x,t) = {b1{x,t),b2{x,t),---,bn{x,t))is the magnetic field, and n(x,t) = p(x,t) + ^\b(x,t)\2 is the total pressure for x € Rn, t > 0. uq(x) and 6o(x) are the initial velocity and initial magnetic field satisfying divuo=O, divbo=O, respectively.In the proof of the continuity of the bilinear form B(u,v), using fractional integral operator, the Lp-boundedness of Hardy-Littlewood maximal operator and the technique of decomposition of the space Rn, we give a succinct and beautiful proof for it. The global existence of mild solutions to the MHD system (0.9) on the space BMO~1(Rn) are obtained for small initial data. Furthermore, the uniqueness of Leray-Hopf weak solutions to the MHD system (0.9) on the space C([0,oo);BMO"1(Rn)) is also proved. In particular, the local existence of solutions to the MHD system (0.9) on bmo~1(Rn) is obtained for small initial data, so does the uniqueness on the space C([0,T);bmo~1(Rn)). Consequently, the well-posedness of the MHD system on more general spaces is showed, and the results about well-posedness are improved. Furthermore, the method of proof for the bilinear estimate in Lemma 3.3.1 of Chapter 3, which is more succinct, is different from that of Koch and Tataru's on Navier-Stokes equations in [50].Part three. We study the incompressible ideal MHD systemwith initial data( u{x,O) ~uo{x),(0.12) [b(x,O)=bo{x).where x € Rn, t > 0, and n > 2 is the space dimension. Hereu(x,t) = (u1{x,t),u2{x,t),---,un{x,t)) is the velocity of the fluid flows, andb(x,t) = {bi(x,t),b2{x,t),---,bn{x,t))is the magnetic field, and n(x,t) = p(x,t) + \\b(x,t)\2 is the total pressure for x € R", t > 0. uo(x) and 6o(x) are the initial velocity and initial magnetic field satisfying divuo=0, div60=0, respectively.In this part we study the incompressible ideal magneto-hydrodynamic system (0.11), and prove the local existence and uniqueness of solutions in critical Besov spaces ￡?p|n//p(Rn) for 1 < p < oo, which improves the previous results.As we all know, for the ideal magneto-hydrodynamic system, to establish its well-posedness is more difficult due to the lack of dissipation. Compared to the case of Euler equations, the properties of some physical quantities are changed because of the couple effect between velocity and magnetic field. For instance, the vorticity is not conserved along particle trajectories even for the two-dimension inviscid fluid flows and magnetic field. So the previous method is not valid. Using Littlewood-Paley dyadic decomposition and Bony's para-product decomposition, we obtain the estimates of bilinear and pressure terms on the Besov space JB^n/p(Rn) and B^{Rn) for 1 < p <00. By means of constructing a sequence of approximating solutions (um(x, t), bm(x, t)), m = \, 2, ■ ? ?, and using the couple effect between velocity and magnetic field fully, we prove that the sequence of approximating solutions (um(x, t),bm(x, ￡)), m = 1, 2,is uniformly bounded on the Besov space Bp*,(]Rn) for 1 < p < oo. Furthermore, we also prove that the sequence of approximating solutions (um(x,t),bm(x,t)), m —1, 2, ? ■ ■ is a Cauchy sequence on the Besov space 5p{p(M") for 1 < p < oo, By-virtue of the partial contraction mapping principle, we prove the local existence and uniqueness of solutions in the critical Besov space Bp^p(W) for 1 < p < oo.Part four. We discuss the blow-up criteria of smooth solutions to the magneto-hydrodynamic system.Firstly, we derive the blow-up criteria on the space BMO by virtue of the couple effect between velocity of fluid and magnetic field. Let (u(x,t),b(x,t)) be a pair of smooth solutions on the interval (0,T). If (u(x,t),b(x,t)) satisfies (u(x,t),b(x,t)) e L2(0,T; BMO), or the vorticity satisfies (rot u(x,t), wt b(x,t)) 6 Ll(0,T;BMO), or the deformation satisfies (Def u(x, t),Def b(x, t)) G Ll(Q,T;BMO), then the solution (u(x, t), b(x,t)) can be extended beyond t = T.Secondly, by means of the logarithmic Sobolev inequality < CH/MII^ log++ jjj^jj^- (0-13)in Besov spaces, where f(x) G B^^W1), s > ~ , p, q G [l,oo], we obtain the blow-up criterion of smooth solutions on the Besov space .￡?￡, (Rn). That isLet (u(x,t),b(x,t)) be the smooth solutions to the MHD system satisfying/ ||(rotu(-,i), rot &(■,*))H^ ^dt < oo. (0.14)J oThen there exists T" > T, such that the smooth solution (u(x, t), b(x, t)) can be continuously extended to the interval (0, T'). Thus, the blow-up criteria of smooth solutions to the Navier-Stokes equations are generalized to the magneto-hydrodynamic system completely.