Existence theory by front-tracking for general nonlinear hyperbolic systems

ABSTRACT. We consider the Cauchy problem for a strictly hyperbolic, NxN quasilinear system in one-space dimension (1) u_t+A(u) u_x=0,, u(0,x) = overline u (x),, where u=u(t,x)=(u_1(t,x),dots, u_N(t,x)), u ---> A(u) is a smooth matrix-valued map, and the initial data overline u is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in L^1_{loc} to the vanishing viscosity solution of (1), which, by the results of Bianchini and Bressan, is the unique limit of solutions to the (artificial) viscous parabolic approximation u_t+A(u) u_x=mu, u_{xx}, u(0,x) = bar u (x),, as mu--> 0. In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in R^N, the limit of front tracking approximations provides a weak solution of the system of conservation laws u_t+F(u)_x=0, satisfying the Liu admissibility conditions. These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), uin R^N. In particular, our construction applies to general strictly hyperbolic system of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine non linearity or of linear degeneracy in the sense of Lax, or in the generalized sense of Liu.