TY - JOUR

T1 - On φ -variation for 1-d scalar conservation laws

AU - Jenssen, Helge Kristian

AU - Ridder, Johanna

N1 - Funding Information:
H. K. Jenssen was partially supported by NSF grant DMS-1813283.
Publisher Copyright:
© 2020 World Scientific Publishing Company.

PY - 2020/12

Y1 - 2020/12

N2 - 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.

AB - 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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U2 - 10.1142/S0219891620500277

DO - 10.1142/S0219891620500277

M3 - Article

AN - SCOPUS:85100533994

SN - 0219-8916

VL - 17

SP - 843

EP - 861

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

IS - 4

ER -