Existence and Uniqueness of the Riemann Problem for a Nonlinear System of Conservation Laws of Mixed Type

We study the system of conservation laws given by \[ \left \{ {_{{\upsilon _t} + {{[\upsilon (a + u)]}_x} = 0\quad (a > 1{\text {is}}{\text {a}}{\text {constant}}),}^{{u_t} + {{[u(1 - \upsilon )]}_x} = 0,}} \right .\] with any Riemann initial data $({u_ \mp },{\upsilon _ \mp })$. The system is elliptic in the domain where ${(\upsilon - u + a - 1)^2} + 4(a - 1)u < 0$ and strictly hyperbolic when ${(\upsilon - u + a - 1)^2} + 4(a - 1)u > 0$. We combine and generalize Lax criterion and Oleinik-Liu criterion to introduce the generalized entropy condition (G.E.C.) by which we can show that the Riemann problem always has a weak solution (any discontinuity satisfies the G.E.C.) for any initial data, however not necessarily unique. We introduce the minimum principle then in the definition of an admissible weak solution for the Riemann problem and the existence and uniqueness of the solution for any Riemann data.