Some Mathematical Theories On Compressible Viscous Radiative And Reactive Gas

Compressible Viscous Radiative and Reactive Gas Equations(CVRRGE),which describes the motions of compressible viscous radiative and reactive gas in the high tem-perature regime,is a fundamental equation in radiation hydrodynamics.The study on its global well-posedness and the precise description of the large-time behaviors of the global solutions obtained to CVRRGE with large initial data is one of the hottest topics in the field of nonlinear partial differential equations in recent years and many excellent results have been obtained.Even so,to the best of our knowledge,all the results obtained so far are concerned with the case when viscosity coefficient is a positive constant and all the results available up to now focus either on the initial-boundary value problem of CVRRGE on bounded domain in one-dimensional space or on the spherically symmetric solutions of CVRRGE on a bounded annular region in high dimensional space.So it is natural to consider the following two types of problems:Problem 1:Since the physical phenomenon described by CVRRGE involve high temperature process and the experimental results for gases at high temperatures in[91]show that the viscosity coeffcient may also depend on density and temperature.Thus a natural and interesting question is how to establish the corresponding glob-al well-posedness theory to CVRRGE with large initial data when the viscosity coefficient depend on the density and temperature?For such a problem,even in the case of ideal polytropic gas,it is still a difficult problem on how to establish a satisfactory global well-posedness theory to the one-dimensional Navier-Stokes equations with temperature-dependent viscosity(see[31]).As far as we know,the only progress in this direction so far is[88],where a global solvability result was obtained for the Cauchy problem of the one-dimensional compressible viscous and heat-conducting ideal polytropic gas with large initial data when the viscosity coefficient depends both on temperature and density.No global solv-ability results are available up to now on the CVRRGE with large data and for density and/or temperature dependent viscosity coefficient;Problem 2:For the case when the viscosity coefficient is a positive constant,how to obtain the corresponding global solvability result and large-time behavior of solutions to CVRRGE with large initial data in unbounded domain?It should be pointed out that even for the case of viscous and heat-conducting ideal polytropic gas,although the corresponding global well-posedness theory to its one-dimensional Cauchy problem has been established for nearly forty years([41,45]),the precise decription of its large-time behavior was obtained only recently([49]).To the best of our knowledge,nothing is known for the Cauchy problem of one-dimensional CVRRGE and spherically symmetric flows to CVRRGE in a n-dimensional exterior domain.This thesis is concerned with the above two types of problems and the main contents include:For the first problem,we first study the global existence of smooth solutions to t-wo types of initial-boundary value problems(free boundary and Dirichlet boundary)to CVRRGE with large initial data when the viscosity coefficient depends on density in Chapter 2.For the free boundary value problem,the viscosity coefficient can be degen-erate,while for the Dirichlet boundary value problem,the viscosity coefficient need to satisfy a non-degenerate condition.Then we study the existence and large-time behav-ior of global solutions for the Cauchy problem of one-dimensional CVRRGE when the viscosity coefficient depends on temperature in Chapter 3.For the second one,we first study the global existence and large-time behavior of solutions for the Cauchy problem of the one-dimensional CVRRGE when the viscosi-ty coefficient is a positive constant in Chapter 4.Then we study the global existence and large-time behavior of spherically symmetric solutions of the initial-boundary value problem to CVRRGE in an exterior domain with large initial data in Chapter 5.The key point to study all the above problems is to deduce the positive lower and upper bounds on both density and temperature.Compared with the mathematical theory of one-dimensional compressible Navier-Stokes equations when all the thermodynamic quantities satisfy the equation of state for ideal polytropic gas,the main difficulties we need to overcome can be outlined as in the following:(1).The first difficulty is how to deduce the upper bound of temperature.For the one-dimensional CVRRGE,due to the fourth-order radiative part in both pressure and the internal energy,the method introduced in[49]to deduce the upper bound of temperature can not be applied.Thus we need to introduce a new method to get the upper bound of temperature.Motivated by[42],we can construct some elaborate auxiliary functions X(t),Y(t),Z(t)and W(t)to deduce the desired upper bound of temperature(see[25,51,55]);(2).The second one is how to deduce the positive lower and upper bounds(time-independent)of density as mentioned in Problem 2.Since the viscosity coefficient,is a positive constant,motivated by[36],we can first deduce a local representation of the density function.Then combining such a representation with the basic energy-estimates,we can obtain the desired bounds;(3).The third one is that,for the Problem 1,since the viscosity coefficient is not a positive constant,we can not obtain the positive lower and upper bounds of density by deducing a local representation as that in Problem 2.Our idea is try to find the relationship between lower and upper bounds of density and temperature by using the underlying structure of CVRRGE under our consideration(for the initial-boundary value problem,we will also use some of the conservation laws induced by the boundary conditions),for example,motivated by[11],we can use the upper bound of temperature to control both the lower and upper bound of density,then we can obtain the desired positive lower and upper bounds on both density and temperature.