Research On The Viscosity Solutions And Weak Solutions To Inhomogeneous Hyperbolic Systems

With the development of fluid dynamics, especially computational fluid dynamics, the study of the system of conservation laws arised in 1950's, it is a very important mathematical models for a variety of physical phenomena. Almost all the continuous mechanics's models are of the form. An important aspect of the theory of the nonlinear hyperbolic system of conservation laws is the existence of solutions. It helps to justify the appropriateness of the models. However, the solutions of the Cauchy problem of the nonlinear hyperbolic system generally develop singularities in finite time even if the initial data are small and smooth.For this reason, solutions must be found in the space of discontinuous functions. Therefore, one can not directly use the classical analytic techniques that predominate in the theory of partial differential equations of other types. Instead, we construct approximate solutions via the singular perturbation methods. By using the compactness of the approximate solutions, we can obtain the existence of the solutions of original system.This dissertation studies the existence of the viscosity solutions and the global weak solutions of the Cauchy problem to systems of inhomogeneous nonlinear hyperbolic conservation laws. The main work of this dissertation is outlined as follows:1. The scheme for the existence of the weak solutions to inhomogeneous conservation laws.Based on the existence theory of homogeneous hyperbolic conservation laws, the general inhomogeneous systems are studied. By using the viscosity method and the compensated compactness theory, the existence and the uniform boundedness of the viscosity solution under suitable conditions are obtained. Therefore there exists a weak convergent subsequence of the viscosity solutions. But in general, weak convergence doesn't imply strong convergence, In order to get the strong compactness of the sequence, suitable entropy-entropy flux pairs are constructed. By using Young measure representation theorems, one just needs to prove that the Young measure derived by viscosity solution sequence is a Dirac measure.2. Existence of global entropy solutions to a non-strictly hyperbolic system with a source. The problem is studied in two cases, the linear source terms and the general source terms. By applying maximum principle we can get the uniform boundedness of the corresponding viscosity solutions; Furthermore we exploit three groups of strong-weak entropy combinations to get the H l?o1 c compactness of the entropy equation; At last, coupling with Kinetic formulation, we can obtain the existence of a strong convergent sequence, then the limit function is just the entropy weak solution of the original systems.3. Existence of global entropy solutions to an inhomogeneous elastodynamics system.For the corresponding viscosity systems, there exists a positively invariant region, prior L∞estimates of vanishing solution is derived with the help of the region. Furthermore, by using a different entropy-entropy flux pair, original results concerned with homogeneous systems are generalized.4. We provide a L∞bounds on the solutions of a semi-linear parabolic system which originated from the theory of phase transitions. The settlement of this problem is of importance for the study of the existence of solutions to the corresponding hyperbolic systems.The method of compensated compactness is widely used in this dissertation. This method is based on the continuity of the weak convergent sequence in functional analysis. Young measure representation theorem, convex entropy and the H l?o1 c compactness of the entropy equation are the key to the method. If the probability measure is a Dirac measure, then the weak convergence implies strong convergence.