Extensions Of Hardy Inequality And Well-posedness For The Magnetohydrodynamics Equations

This dissertation is devoted to extend Hardy inequality by using harmonic analysis and study the well-poseness of Magnetohydrodynamics equation by Littlewood-Paley decomposition.Frist, we introduce some elementary tools in the first chapter for convenience. Among these tools, someone are well-known, but we list them which are given simple proof or are only given results.Next , in the second chapter , we prove sharp Hardy inequalities by using Maximal function theory. Our results improve and extend the well-known results of G.Hardy [1], T.Cazenave [2], J.-Y Chemin[4] and T.Tao[6].Finally , we study the well-posedness of incompressible MHD equation by Bony decomposition.Magneto-hydro-dynamics(MHD) equation is a equation which explains fluid dynamics of a plasma which can carry an electric current in magnetic field. It's well known that the question whether Navier-Stokes equation has global regularity solution is very hard. In other word, whether the regularity of the weak solution of NS equation with initial data in L²(R³) can be improved? The problem is very hard and open since it's a supercritical problem. Therefore, we consider a easier problem with initial data in critical space. A same problem for MHD equation attract many authors' interest as well as NS equation. In this charter, we devote to consider the well-posedness of MHD in a critical Besov space who degree is -1. Comparing to Kato's space [3] for Navier-Stokes equation, we give existence and uniqueness of the solution of MHD in L^q([0,T];B_p,t^{(3/p)+(2/q)-1}) with (p,q,r)∈[1,2[×[2,∞] r [1,∞] such that 3/p + 4/q>2 or (p, q, r)∈[2,∞]²Ã—[1,∞] such that 3/p+2/q> 1 by applying contraction argument directly.