Existence theory by front tracking for general nonlinear hyperbolic systems

We consider the Cauchy problem for a strictly hyperbolic, N x N quasilinear system in one-space dimension ut + A(u) ux = 0, u(0, x) = ū(x), (1) where u = u(t,x) = (u1(t, x),..., uN(t, x)), u → A(u) is a smooth matrix-valued map and the initial data u is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in L1loc to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial) viscous parabolic approximation ut + A(u) ux = μ uxx, u(0, x) = u(x), as μ → 0. In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in RN, the limit of front tracking approximations provides a weak solution of the system of conservation laws ut + 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), u 1∈ RN. In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of LAX [17], or in the generalized sense of Liu [23].