A Class of Nonlinear, Nonhyperbolic Systems of Conservation Laws with Well-Posed Initial Value Problems

With mild restrictions on the initial data, we show well-posedness of the initial value problem for systems of conservation laws in one space variable with real and equal characteristic speeds and a deficiency of one corresponding eigenvector. The obtained weak solutions assume values as Borel measures at each time after a smooth solution ceases to exist. The concept of entropy density–flux functions is extended to such systems. Finally, we show that systems with well-posed initial value problems, admitting weak solutions of this type, necessarily satisfy these structural assumptions.