The Littlewood-paley, Theory And Its Applications In The Hydrodynamic Equations

Since the 80's,Fourier analysis methods have known a growing interest in the study of linear and nonlinear PDE's.In particular,techniques based on Littlewood-Paley decomposition and paradifferential calculus have been proved to be very efficient.Littlewood-Paley decomposition has been introduced more than fifty years ago in harmonic analysis but its systematic use in the PDE's framework is rather recent.Paradifferential calculus,as for it,has been introduced in 1981 by J.-M.Bony for the study of the propagation of microlocal singularities in nonlinear hyperbolic PDE's(see[7]).In this thesis,we use theory of Littlewood-Paley decomposition and Bony's paraproduct decomposition to study several types of hydrodynamic equations: Navier-Stokes equations,Magneto-Micropolar fluid equations,generalized Magnetohydrodynamics equations,dissipative Quasi-Geostrophic equations and generalized Camassa-Holm equations.We study their basic properties such as the local existence,the regularity of the weak solutions and the blow-up criteria of strong solutions etc.In the second chapter,we recall the theory of Littlewood-Paley.We introduce the Littlewood-Paley decomposition,by which we define Besov space and time-space Besov space.Then we introduce Bony's paraproduct decomposition technique.Some properties and estimates of Besov space and Bony's paraproduct are listed.In the third chapter,we study Navier-Stokes equations.Navier-Stokes equations are the basic equations describing the movement of the fluid,the problem of global existence is one of the open problem attracting many mathematicians.For the two dimensional case,corresponding problems have been solved.We consider n-dimension(n≥3) Navier-Stokes equations and set up the blow-up criteria of strong solutions and the regularity of the weak solutions.In the fourth chapter,by using of some harmonic methods including Bony's paraproduct decomposition,commutator estimate based on frequency localization, high-low frequency decomposition,we impose detailed analysis on the dissipative term and the nonlinear term of the three dimensional Magneto-Micropolar fluid equations,set up a series of more refined estimates,we then obtain the wellposedness of the smooth solution.In the fifth chapter,using Littlewood-Paley decomposition and paradifferential calculus,we study local well-posedness and the regularity of the weak solutions of the generalized Magneto-hydrodynamics equations,refined some results.In the sixth chapter,we consider the 2D dissipative quasi-geostrophic equations and study the regularity criterion of the solutions.By means of a commutator estimate based on frequency localization and Bony's paraproduct decomposition, we obtain a regularity criterion which improves the result of Dong and Chen[43].In the last chapter,we know that,after simple transform,Camassa-Holm equations have many similarity with transport equations.Using estimates of transform equations in the frame of Besov spaces,compactness arguments and Littilewood-Paley theory,we obtain local wellposedness of generalized CamassaHolm equations.