On the theory of generalized entropy solutions of the Cauchy problem for a class of non-strictly hyperbolic systems of conservation laws

Many-dimensional non-strictly hyperbolic systems of conservation laws with a radially degenerate flux function are considered. For such systems the set of entropies is defined and described, the concept of generalized entropy solution of the Cauchy problem is introduced, and the properties of generalized entropy solutions are studied. The class of strong generalized entropy solutions is distinguished, in which the Cauchy problem in question is uniquely soluble. A condition on the initial data is described that ensures that the generalized entropy solution is strong and therefore unique. Under this condition the convergence of the “vanishing viscosity” method is established. An example presented in the paper shows that a generalized entropy solution is not necessarily unique in the general case.