Sufficient conditions for local existence via Glimm's scheme for large BV data

Abstract

Using Glimm's scheme, sufficient conditions are derived for the global existence of a weak solution to a strictly hyperbolic genuinely nonlinear system of partial differential equations in one space dimension when the initial data is a small BV perturbation of a solvable Riemann problem. By using the finite propagation speed of the system, this yields a local existence theorem for arbitrary BV initial data that satisfies the above-mentioned conditions at all large jumps. The case of linearly degenerate fields is also treated, and the results are applied to the p-system and to the 1-D nonisentropic γ-gas-law Euler equations.