Nonlinear hyperbolic systems: Non-degenerate flux, inner speed variation,and graph solutions

Olivier Glass (1) , Philippe G. LeFloch (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We study the Cauchy problem for general, nonlinear, strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves, only; we establish thatthe ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third,we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time-regularity of the graph solutions (X, U ) introduced by the second author, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.

Domains: Numerical Analysis [math.NA]

Olivier Glass, Philippe G. LeFloch. Nonlinear hyperbolic systems: Non-degenerate flux, inner speed variation,and graph solutions. Archive for Rational Mechanics and Analysis, 2007, 185, pp.409-480. ⟨hal-00121717⟩