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with initial values f (u+, u+), x > 0, M(z,0)= (1.2) I (u ,v ), x < 0. System (1.1) has the two eigenvalues Ai = u and A2 = 2u with corresponding right eigenvectors r\ = (0,1),r2 = (1 ,v/u)T, and VAi • r 1 =0, VA2 • r2 = 2. Thus (1.1) is nonstrictly hyperbolic and Ai is linearly degenerate; A2 is genuinely nonlinear. We recall that the classical Riemann solution is composed of a shock (contact discontinuity) or rarefaction of the slower family followed by a wave of the second. But in the present situation, one finds that no classical weak solutions exist for certain states (u±,v±), and distributions of the form of Dirac delta shock waves supported on discontinuity lines are found necessary, even though for systems of conservation laws satisfying the classical assumptions (strict hyperbolicity and genuine nonlinearity), the Riemann problem breaks down for some large data [5]. For the two-dimensional system of conservation laws we refer the reader to [10]. System (1.1) with trivial difference (t —> 21) was studied by Korchinski [6] in his Ph.D. Thesis in 1977. Generalized delta-functions were used in his numerical study and in the construction of his unique solution to the Riemann problem. Afterwards, Tan, Zhang, and Zheng [11] introduced a viscosity term in the first equation of (1.1),

A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws