The mixed initial–boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics

Abstract

In this paper, we investigate the mixed initial–boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space { ( t , x ) | t ≥ 0 , x ≥ 0 } . As the result in [A. Bressan, Contractive metrics 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 data is large, we prove that, if the C 1 norm of the initial and boundary data is bounded but possibly large, and the BV norm of the initial and boundary data is sufficiently small, then the solution remains C 1 globally in time. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space–time R 1 + n , are also given.