Research On The Global Solutions Of A Kind Of Viscously Hyperbolic Systems With Relaxation Terms

Systems of conservation laws are very important mathematical models for a variety of physical phenomena appeared in 1950s. Almost all the continuous mechanical models belong to the form.Many people have done research on the existence of solutions for the systems of nonlinear hyperbolic and the problem of relaxation limit for the hyperbolic conservation laws, which are from some phenomenon of physical situation, such as traffic-flow, isentropic gas dynamics, and so on.By applying the method of vanishing artificial viscosity and the compensated compactness, we discussed the singular limit for a kind of nonlinear hyperbolic systems with relaxation and diffusion, that is the convergence of solutions when the relaxation timeτtends to zero faster than the diffusion parameterε,τ= o(ε),ε→0. The main work of this paper is as follows:1. Studied the convergence of solutions for a kind of traffic-flow model with nonlinear terms and relaxation terms. We construct approximate solutions via the singular perturbation methods. First, we get a local solution by using the general contracted mapping principal and the Lebesgue dominated convergence theorem. Then, by applying the extremum principle, we constructed an invariant region, so that we get an a priori bound that is uniformly with respect toεandτfor the global solutions. Further more, whenτ= o(ε),ε→0, we get the entropy pair (η, q) of equilibrium equation satisfyingη(ρ^ε,τ ) t + q(ρ^ε,τ)_x compacted in H l?o1 c.2. Studied the relaxation limit for the system of isentropic gas dynamics. The key is the construction of the invariant region and the compactness of entropy.3. Studied the singular limit for the system of elasticity with relaxation and diffusion.Most of the results in this dissertation are presented by means of compensated compactness. This method is based on the continuity of the weak convergent sequence in functional analysis. Compactness of the entropy equation is the key to the method. If the probability measure is a Dirac measure, then the weak convergence become strong convergence.