On φ -variation for 1-d scalar conservation laws

Let φ: [0,∞) → [0,∞) be a convex function satisfying φ(0) = 0, φ(x) > 0 for x > 0, and limx 0φ(x) x = 0. Consider the unique entropy admissible (i.e. Kružkov) solution u(t,x) of the scalar, 1-d Cauchy problem ∂tu(t,x) + ∂x[f(u(t,x))] = 0, u(0) = u. For compactly supported data u with bounded φ-variation, we realize the solution u(t,x) as a limit of front-tracking approximations and show that the φ-variation of (the right continuous version of) u(t,x) is non-increasing in time. We also establish the natural time-continuity estimate ∫ℝφ(|u(t,x) - u(s,x)|)dx ≤ C · φ -varu(s) ·|t - s| for s,t ≥ 0, where C depends on f. Finally, according to a theorem of Goffman-Moran-Waterman, any regulated function of compact support has bounded φ-variation for some φ. As a corollary we thus have: if u is a regulated function, so is u(t) for all t > 0.