The Semigroup Approach to Conservation Laws with Discontinuous Flux

Abstract

The model one-dimensional conservation law with discontinuous spatially heterogeneous flux isWe prove well-posedness for the Cauchy problem for (EvPb) in the framework of solutions satisfying the so-called adapted entropy inequalities.Exploiting the notion of integral solution that comes from the nonlinear semigroup theory, we propose a way to circumvent the use of strong interface traces for the evolution problem (EvPb) (in fact, proving existence of such traces for the case of x-dependent f l,r would be a delicate technical issue). The difficulty is shifted to the study of the associated one-dimensional stationary problem u + f(x, u)x = g, where existence of strong interface traces of entropy solutions is an easy fact. We give a direct proof of this fact, avoiding the subtle arguments of kinetic formulation [23] or of the H-measure approach [27].

Weak asymptotic methods for scalar equations and systems

“…We assume f and g have same sign and that g is globally bounded in sup norm on R. Then Theorem 8. Provided H(x+ , )−H(x, ) tends to +∞ slowly enough when → 0 then the solutions of the ODE (10,11,86) provide a weak asymptotic solution to equation (9).…”

“…Existence. Under assumption i) we construct, as solutions of the ODEs (10,11), a family of functions (x, t −→ u(x, t, )) , T n × [0, +∞[ −→ R, 0 < < η small enough, which, for fixed , are of class C 1 in t and of class L ∞ in x ∈ T n and tend to satisfy equation (1) …”

“…We choose c(t) increasing with t. We remark that the changes of f ± into c(t) + f ± do not modify the proof of weak asymptotic solution (based on f = f + + f − ) and even the solution of the ODE (10,11) in the sense that c(t) brings no more than a vanishing viscosity, as this is noticed in [32] section 3.…”

“…Such t 0 , x 0 exists from the assumption that u 0 (., 0 ) is continuous: then the solution u(x, t, 0 ) is continuous since the ODEs (10,11) can be considered in the…”

“…Let us recall that when f depends on x the solution can be unbounded (6), but the presence of t would not cause problem in this section. (10,11) takes its values in between min x∈T u 0 (x, ) and max x∈T u 0 (x, ).…”