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

We consider a hyperbolic system of conservation laws on the space of symmetric second-order matrices. The right-hand side of this system contains the functional calculus operator $\tilde f(U)$generated in the general case only by a continuous scalar function $f(u)$. For these systems we define and describe the set of singular entropies, introduce the concept of generalized entropy solutions of the corresponding Cauchy problem, and investigate the properties of generalized entropy solutions. We define the class of strong generalized entropy solutions, in which the Cauchy problem has precisely one solution. We suggest a condition on the initial data under which any generalized entropy solution is strong, which implies its uniqueness. Under this condition we establish that the “vanishing viscosity” method converges. An example shows that in the general case there can be more than one generalized entropy solution.