Researches On Global Wellposedness Of Nonlinear Evolutionary System

The basic mathematical models are partial differential equations in many important physical,mechanical discipline.Partial differential equations is the most active branch of mathematics is one of many elements of mathematics and physics,the foundation helps people right from micro to macro-laws of physical movement to understand the basic laws of natural science are mostly accurate mathematical expression differential equations.This paper discusses a class of profound physical meaning of the Navier-Stokes equations and Thermo-visco-elastic equation;we use some new techniques,ideas and methods.Studied the solution of these equations as a whole existence,self-similar solutions and the overall energy of the explosion,the inner solution regularity,energy issues and the equation of exponential decay of the same compact attractor problem.Slightly rough speaking,we have studied the following issues:●In chapter two,we concerned with the self-similar solutions to the compressible Navier-Stokes equations of a 1D viscous polytropic ideal gas.Our results show that there exist neither forward nor backward self similar solutions with finite total energy,which generalizes the results for the case of the isothermal compressible Navier-Stokes equations in Z.Guo and S.Jiang(Self-similar solutions to the isothermal compressible Navier-Stokes).●In chapter three,we shall establish some global existence results of a 3D hyperbolic system arising in the Green-Naghdi models of thermoelasticity of typeⅡwith a dissipative boundary condition for the displacement.The existence and exponential decay of energy of linear problem has been solved by Lazzari and Nibbi.Furthermore,we shall establish the global existence of semilinear and nonlinear thermoelastic systems by using semigroup approach.●In chapter four,we concerned with the exponential decay of energy for a dissipative multi-dimensional nonhomogeneous and anisotropic elastic system with a locally reacting boundary subject to small oscillations.Under reasonable assumptions,multiplicative techniques and energy method are used.●In chapter five,we prove the regularity,exponential stability of global solutions and existence of uniform compact attractors of semi processes,generated by the global solutions, of a two-parameter family of operators for a nonlinear one-dimensional non-autonomous equation of viscoelasticity.We employ the properties of the analytic semi group to show the compactness for the semi process generated by the global solutions.