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.


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Work Title On φ -variation for 1-d scalar conservation laws
Open Access
  1. Helge Kristian Jenssen
  2. Johanna Ridder
License In Copyright (Rights Reserved)
Work Type Article
  1. Journal of Hyperbolic Differential Equations
Publication Date December 1, 2020
Publisher Identifier (DOI)
  1. https://doi.org/10.1142/S0219891620500277
Deposited August 20, 2021




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