Existence result for a class of nonconservative and nonstrictly hyperbolic systems

We consider the class of nonconservative hyperbolic systems partial derivative(t)u+A(u) partial derivative(x) u = 0, partial derivative(t)upsilon + A(u) partial derivative(x) upsilon = 0, where, mu = u(x, t), upsilon = upsilon(x, t) is an element of IRN are the unknowns and A(mu) is a strictly hyperbolic matrix. Relying on the notion of weak solution proposed by Dal Maso, LeFloch, and Murat ("Definition and weak stability of nonconservative products", J. Math. Pures Appl. 74 (1995), 483-548), we establish the existence of weak solutions for the corresponding Cauchy problem, in the class of bounded functions with bounded variation. The main steps in our proof are as follows: (i) We solve the Riemann problem based on a prescribed family of paths. (ii) We derive a uniform bound on the total variation of corresponding wave-front tracking approximations mu(h), upsilon(h). (iii) We justify rigorously the passage. to the limit in the nonconservative product A(mu(4))partial derivative(x)upsilon(h), based on the local uniform convergence properties of mu(h), by extending an argument due to LeFloch and Liu ("Existence theory for nonlinear hyperbolic systems in nonconservative form", Forum Math. 5 (1993), 261-280). Our results provide a generalization to the existence theorem established earlier in the scalar case (N = 1) by the second author ("An existence and uniqueness result for two nonstrictly hyperbolic systems", IMA Volumes in Math. and its Appl. 27,"Nonlinear evolution equations that change type"