Overall Posedness Anisotropy Boussinesq Equation

The fluid dynamic equation (system) as the basic model to describe fluid flow, is an important nonlinear partial differential equations to understand the natural phe-nomena. It lies in the hot research areas of mathematics and physics. For instance, Boussinesq equations describe the two distinctive features in the dynamics of the ocean or of the atmosphere:rotation and stratification. This system have been intensively studied due to their physical background and mathematical significance.Great progress about Boussinesq system has been made in the past years, espe-cially the study for the two-dimensional subcritical or critical Boussinesq equations is in a satisfactory state. Recently, by establishing the growth estimate of the velocity field and using the structure of equations, the global well-posedness for the two-dimensional anisotropic Boussinesq system has been achieved. However, the uniqueness of weak so-lution or the global well-posedness of smooth solution for the tridimensional Boussinesq equations is still an open problem. To better understand the influence of the convec-tion term in the fluid, some researchers consider some fluids with special structure (for example:axisymmetric flow without swirl).This thesis is devoted to the study of Cauchy problem for the anisotropic Boussi-nesq equations. By using the coupling structure of Boussinesq equations, properties of axisymmetric flow and losing estimates for the non-Lipschitz vector field, we establish some global results on tridimensional anisotropic Boussinesq system under assumption that the initial data is axisymmetric without swirl; by using Fourier localization method and the growth estimate of the velocity, we prove the global well-posedness of the two-dimensional anisotropic non linear Boussinesq system for the rough initial data. In addition, we are concerned with some else relevant models. We investigate some prob-lems such as the regularity criterion of tridimensional incompressible Navier-Stokes equations、the global well-posedness for the high dimensional compressible Navier-Stokes-Poisson equations in the LP framework、 the ill-posedness to Keller-Segel system with fraction diffusion.The detail of this thesis is arranged as follows.In the second chapter, we recall the basic Littlewood-Paley theory. Next, we give some useful lemmas and a commutator estimate. At last we review algebraic and geometric properties of axisymmetric flow.In the third chapter, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is axisymmetric without swirl, by using structure of the coupling of Boussinesq equations and the horizontal smoothing effect, we prove the global well-posedness for this system. In our proof, the main ingredient is to establish a magic relationship between yr/r and wθ/r by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis.In the fourth chapter, we continue to consider Cauchy problem of the tridimen-sional anisotropic Boussinesq equations. Under the assumption that the support of the axisymmetric initial data po(r,z) does not intersect with the axis (Oz), we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity p/r for large time by taking advantage of characteristic of trans-port equation. This growing property together with the horizontal smoothing effect enables us to establish the estimate for the quantity which implies However, space L.2admits forbidden singularity to prevent us from getting the high regularity of (ρ,u). To bridge this gap, we exploit the space-time estimate about by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space-time logarithmic inequality.In the fifth chapter, we investigate the regularity criterion of the tridimensional Navier-Stokes equations via one velocity component. By deeply using properties of the incompressible fluid, we mainly establish regularity criteria of Leray-Hopf weak solutions via only one element Λriuj with γ∈[0,1] and i, j∈{1,2,3}, that is Here F is the set of index (α,β) which appears in our results and the fractional operator This extends and improves some known regularity criterions of Leray-Hopf weak solutions in term of one velocity component, including the notable works of C. Cao and E. S. Titi (Arch. Ration. Mech. Anal.202(2011)919-932). More importantly, by making full use of the Bony paraproduct decomposition, we show that Leray-Hopf weak solutions is smooth on [0, T] in term of (at the endpoint a=∞) where i,j∈{1,2,3}.In the sixth chapter, we firstly study the global well-posedness for the two dimen-sional non Boussinesq equations with vertical dissipation. Next, we consider a global well-posedness of the compressible Navier-Stokes-Possion equations in IP framework with small initial data. Finally, we prove the ill-posedness of Keller-Segel equations with fractional diffusion in critical Fourier-Herz space.