Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data

Abstract

We investigate the existence of a global classical solution to the Goursatproblem forlinearly degenerate quasilinear hyperbolic systems of diagonal form. As the result in [A. Bressan, Contractivemetrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409-421]suggests that one may achieve global smoothness even if the $C^1$ norm of the initial datais large, we prove that, if the $C^1$ normand the BV norm of the boundary data are bounded but possiblylarge, thenthe solution remains $C^1$ globally in time. Applications include theequation of time-like extremal surfaces in Minkowski space$R^{1+(1+n)}$ and the one-dimensional Chaplygin gas equations.