Existence of the Entropy Solution for a Viscoelastic Model

In this paper, we consider the Cauchy problem for a viscoelastic model with relaxation with discontinuous, large initial data, where 0<μ<1,δ>0 are constants. When the system is nonstrictly hyperbolic, under the additional assumptionv0x L∞, the system is reduced to an inhomogeneous scalar balance law by employing the special form of the system itself. After introducing a definition of entropy solutions to the system, we prove the existence, uniqueness, and continuous dependence of the global entropy solution for the system. When the system is strictly hyperbolic, some special entropy pairs of the Lax type are constructed, in which the progression terms are functions of a single variable, and the necessary estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations. The special entropy pairs are used to prove the existence of the global entropy solutions for the corresponding Cauchy problem by applying the method of compensated compactness