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.