ABSTRACT
Systems of hyperbolic conservation laws are very important mathematical models for a variety of physical phenomena that appear in traffic flow, theory of elasticity, gas dynamics, fluid dynamics and so on. In general, the classical solution of the Cauchy problem for nonlinear hyperbolic conservation laws exists only locally in time even if the initial data are small and smooth. This means that shock waves always appear in the solution for a suitable large time. An important aspect of the theory of nonlinear system of conservation laws is the question of existence of solutions to these equations. It helps to answer the question if the modelling of the natural phenomena at hand has been done correctly, and if the problem is well posed. This chapter introduces some applications of the method of compensated compactness to the scalar conservation law as well as some special systems of two or three equations. It considers applications to physical systems with a relaxation perturbation parameter.
