Spatial Decay Estimates And Structural Stability Of Some Partial Differential Equations

In this dissertation,we mainly consider the spatial decay estimates of Boussinesq equations and the structural stability of several kinds of primitive equations.The first problem we study is a class of Quasilinear parabolic equations simulating the concentration diffusion of Newtonian fluid through porous media.We obtain the global existence of solutions for the system of equations by limiting the parameters in the equations.When the solutions blow up,the lower bound of blow-up time is obtainedThe second problem we study is the spatial decay of Boussinesq equations in a semi-infinite pipe.We consider the Boussinesq equations,in which the heat conduction coefficient depends on the temperature,and transform it properly.Finally,by introducing a stream function,defining an energy expression,using the relevant achievements in the current literature and the method of energy estimation,a differential-integral inequality is obtained.The spatial decay estimates of energy are obtained by solving the inequality.In order to make our research meaningful,we derive the explicit upper bound of total energy.The third problem we study is the structural stability of the primitive equations.The primitive equations are the basic models to study the ocean and atmosphere movement,which are used to understand the mechanism of weather forecast and climate change.When there is saturated steam,the following primitive equations of moist atmosphere was given as(?)We firstly derive a priori bounds of the solutions.Secondly,we assume that there are two solutions corresponding to different heat sources,water vapor sources or boundary parameters of the equations.Finally,we prove that when the two different heat sources,water vapor sources or boundary parameters are infinitely close,the two solutions of the equations are infinitely close.The continuous dependence of the equations is obtained.Then,we consider large-scale ocean motion.In the large-scale ocean motion,the curvature of the earth is not considered,so we can use the Cartesian coordinates instead of the spherical coordinates.The three coordinate axes under the Cartesian rectangular coordinate system are represented by ox,oy,oz:ox:east-west direction,oy:north-south direction,oz:vertical direction.Inspired by relevant references,we assume that all unknown functions of the primitive equations are independent of y and the function v is non-zero.We first introduce the relevant results of the two-dimensional primitive equations,and transform the primitive equations.Secondly,we derive the upper bounds of the various norms.Finally,we prove the convergence of the equations to the boundary parameters.