Weak Solution And 0-Relaxation Limits Of Inhomogeneous Hyperbolic Systems

Systems of hyperbolic conservation laws are very comment equations which are usually appeared in physics, chemical and biographic. They are very important mathematical models for a variety of physical phenomena, almost all the continuous mechanics'models belong to the form. But they are also special ones, especially the case of nonlinear, the solutions of Cauchy problem may fail to be continuous even if the initial data are smooth. So we can't research them in the continuous functions spaces, neither use some tools, like functional analysis, which are powerful to other kinds of equations. In view of this, we introduce weak solutions to extend the range of solutions, and then combine the viscosity vanishing method and the theory of compensated compactness to research the corresponding viscosity solutions, then convergence to the weak solutions.By combination of the viscosity vanishing method and the theory of compensated compactness, we mainly discuss the solutions of systems of two equations with inhomogeneous terms and the stiff relaxation terms in this article, the main contents included are as following:1. The viscosity solutions and the existence of weak solutions of general inhomogeneous conservation laws.On the basic of the heat equation theory, we mainly use the iterative method to discuss the global existence of solutions of conservation laws with general inhomogeneous terms and property of the solutions, such as the boundedness, continuous, and the viscosity solutions convergence to the weak solutions by the compensated compactness.2. 0-relaxation limits of conservation laws with stiff relaxation terms.In this section, under the assumption that the viscosity solutions are bounded, we get the convergence of the 0-relaxation limits of the conservation laws with general inhomogeneous terms and relaxation terms, by the theory of compensated compactness; and then, we apply the conclusion to some concrete systems, our work is focus on the validating the boundedness via the theory of positive invariant region and the conditions of the theorem of weak limits.