Global Solutions To Some Dissipative Nonlinear Hyperbolic Conservation Laws With Large Initial Data

This thesis is concerned with the global solvability and the precise description of large time behaviors of global solutions to some dissipative nonlinear hyperbolic conser-vation laws with large data. It consists in the following two parts:The first part focuses on the global nonlinear stability of strong planar rarefaction waves for the Jin-Xin relaxation approximation of scalar conservation laws in several spatial dimensions. The study on the nonlinear stability of elementary waves to dissi-pative nonlinear hyperbolic conservation laws is one of the hottest topic in the field of nonlinear partial differential equations and the key point to this problem lies in how to control the possible growth of its global solutions caused by the nonlinearity of the system under consideration and/or the interactions of waves from different families.. An effective well-developed argument is to employ the smallness of both the initial perturba-tion and the strength of the rarefaction waves to control the possible growth mentioned above. And based on such an argument, the local stability of weak basic wave patterns to some dissipative fluid dynamical systems are well-established, but the problem on how to deduce the corresponding global stability results for strong elementary waves remains open. One purpose of our thesis is concerned with the global nonlinear stability of strong planar rarefaction waves for a typical dissipative nonlinear hyperbolic conser-vation laws, i.e., the so-called Jin-Xin relaxation approximation to scalar conservation laws in several space dimensions. For such a problem, since we need not to deal with the problem induced by the interactions of waves from different families, all that we needed to control is the possible growth of the solutions caused by the nonlinearity of the sys-tem under consideration. Such a problem has been studied by many people and local stability of weak planar rarefaction waves and local stability of strong planar rarefaction waves have been obtained by Tao Luo in1997[54] and Huijiang Zhao [87] in2000, but up to now, no result has been proved for global stability of strong planar rarefaction waves. By utilizing the underline structure of the system under consideration and by employing the continuation argument, three types of results on the global stability of strong planar rarefaction waves are obtained.The second part deals with the global smooth solvability of the one-dimensional compressible Navier-Stokes equations with degenerate density and temperature depen-dent transport coefficients. The case when the viscous coefficient μ and heat conductivity coefficient κ, are positive coefficients and the gas is ideal polytropic has been studied by some Russian mathematicians such as Y. Kanel [37], A. Kazhikov [40], etc.. But if we derive the compressible Navier-Stokes equations from the Boltzmann through the Chapman-Enskog expansion, these transport coefficients must be functions of the abso-lute temperature and consequently a more important and more interesting problem is to study the construction of global smooth non-vacuum solutions to the one-dimensional compressible Navier-Stokes equations with temperature and density dependent trans-port coefficients and large data. But compared with the case when the transport coeffi-cients are positive constants, the dependence of viscous coefficient and heat conductivity coefficient, especially the dependence of the viscous coefficient on the absolute temper-ature, has turned out to be especially problematic and as far as we know, there is no global solvability results with large initial data for such a case up to now. Notice that for a class of solid-like material, cf.[9], and some experiments for gases at very high temperature, cf.[39],the viscous coefficient μ may depends on density and the heat con-ductivity may depend on both density and absolute temperature and one has been able to incorporate various forms of density dependence in μ and also density and tempera-ture dependence in κ, cf.[9],[33],[39],[72]. Even so, as we know, in all these works, although the viscous coefficients may depend on density and the heat conductivity may depend on density and temperature, the basic idea used in these manuscripts is first to the positive lower and upper bounds on the density based on the assumptions on the transport coefficients and the state equations for the gases under consideration, then the maximum principle can deduce the lower bound for the absolute temperature and finally to deduce the upper bound for the temperature. If we focus on the ideal polytropic gas, to guarantee the validity of such an argument, generally speaking, the viscous coefficient is assumed to be either a positive constant (in such a case the heat conductivity can depend on both density and temperature, cf.[33],[39],[72], or a fuction of the density but ask that the heat conductivity coefficient is a function of density and temperature with positive lower bound (in some cases the viscosity coefficient is also assumed to have lower and upper bounds), cf.[9],[39].In this thesis, what we are interested in is the construction of global smooth so- lutions away from vacuum to the one-dimensional non-isentropic compressible Navier-Stokes equations with degenerate density and temperature dependent transport coeffi-cients with large initial data. Here degenerate means that as density and temperature tend to zero, the transport coefficients tend also to zero. The key point to this problem is to deduce the positive lower and upper bounds on the density and absolute tempera-ture. By making use of the structure of the one-dimensional compressible Navier-Stokes equations, we found that the lower of the absolute temperature can be bounded by the upper bound of the density. Based on such an observation and by exploiting the argument developed by Y. Kanel in [37], we can obtained the following three types of results:●When the viscous coefficient depends on density, the heat conductivity coefficientdepends on both density and temperature, and both of them are degenerate func-tions of its arguments, two global solvability results are obtained for the Cauchy problem;●When the viscous coefficient depends on density, the heat conductivity coefficient de-pends on both density and temperature, and both of them are degenerate functions of its arguments, three global solvability results are obtained for the corresponding three types of initial-boundary value problem;●For the case when transport coefficients are degenerate functions of the absolute temperature, a Nishida-Smoller type result, cf.[66], is obtained provided that the adiabatic exponent is sufficiently close to1.