The Behaviors For Equations Modelling Fluid Dynamics

The mathematical models of fluid dynamics are described by the mass conservation, momentum conservation, energy conservation together with the basic laws of thermody-namics. They play an more and more important role in theoretical and applied studies in meteorological, atmospheric and oceanographic sciences and petroleum industries. Since Reynolds gave the famous experiments on turbulence in1883and Leray derived the exis-tence of weak solutions of viscous incompressible fluid flows in1930;s, the mathematical theory in fluid dynamics has been attracted more and more attention [1-106], especially in the fields of partial differential equations and nonlinear sciences. For the3D equa-tions modelling fluid dynamics, the fundamental issues which have longstanding interest to mathematicians is the well-posedness and large time behavior of the solutions. This paper is mainly devoted to studying the behaviors of some classic equations modelling fluid dynamics. More precisely, we are focused on the regularity criteria and stability of solutions to some nonlinear partial differential equations in fluid dynamics. The main results are made in the chapter2-chapter6.In Chapter1, we first recall some fundamental theory of the nonlinear partial differ-ential equations and some well-known results of the classic Navier-Stokes equations such as existence, uniqueness and regularity of weak solutions.In Chapter2, we study the regularity criterion for weak solution of three-dimensional magnetohydrodynamic equations in the critical Besov space B∞,∞(R3). Our proof here is divided into two cases:r=1and-1<r<1. For the first case, we use Littlewood-Paley decomposition methods and decompose the nonlinear terms into two parts:low frequency and high frequency. For the second one, we use the Fourier localization technique and the Bony’s para-product decomposition.In Chapter3we are focused on the blow-up criterion of three-dimensional Boussinesq equations with zero diffusion in the largest critical Besov space. Since the absence of the thermal diffusivity, the methods in previous results where both the kinematic viscosity and the thermal diffusivity play a central role can not work for our case here any more. In order to come over this difficulty, we first apply the Bony’s decomposition and Littlewood-Paley decomposition to the momentum equations only and derive the uniform H1estimate of the velocity field. And then, with the aid of the elliptic regularity theory, the energy estimates of solutions, the commutation estimates and some technical derivations, we obtain the uniform Hm estimates of the smooth solutions (v,θ).In Chapter4, we consider the regularity criteria for Cauchy problem of the3D mi-cropolar fluid flows. Firstly we examined the local LP strong solutions with the aid of LP-Lq estimates of the linear equations and the Banach contraction principle. Then we obtained some regularity criteria of the weak solutionf under critical growth conditions, which are only applied on the pressure field in some critical spaces such as Morrey, BMO and Besov spaces. Since the micropolar model actually exhibits asymmetry tress tensor and the non-divergence free micro-rotation vector field. This leads to an additional dif-ficulty in deriving a priori estimates on both the velocity field v and micro-rotation field w. More rigorous analysis is required to deal with the asymmetry stress tensor parts and the nonlinear convection terms. We make full use of the Littlewood-Paley decomposition to divide the nonlinear terms into three parts:low frequency, middle frequency and high frequency. In the three part of this chapter, we devoted to the Logarithmical regularity criteria of three-dimensional micropolar fluid equations in terms of the pressure. The rig-orous analysis due to the new structure in nonlinear terms of the system is made and some anisotropic function inequalities are also employed.In Chapter5, we investigate the stability behaviors for weak solutions of three-dimensional Navier-Stokes equations. With the aid of the energy-type equality and the continuity of the trilinear form∫R3(v·▽v) w dxdt, we first examined the uniformly sta-bility of weak solutions under the small perturbation. Secondly, we showed the asymptotic stability of weak solutions under the large initial and external forcing perturbation. Ad-ditional, energy equality and weak-strong uniqueness for three-dimensional Navier-Stokes equations are also obtained.Chapter6is devoted to the study of the stability issue of the supercritical dissipative surface quasi-geostrophic equation with nondecay low-regular external force in the BMO space and Morrey space. Firstly we showed the uniform stability of the solution for the surface quasi-geostrophic equation in critical BMO space with the aid of energy method. Then we investigate the optimal convergence rate of the Supercritical quasi-geostrophic in the critical Morrey space.