Regularity estimates for continuous solutions of α-convex balance laws

Laura Caravenna

doi:10.3934/cpaa.2017031

Article Contents

2017, Volume 16, Issue 2: 629-644. Doi: 10.3934/cpaa.2017031

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Regularity estimates for continuous solutions of α-convex balance laws

Laura Caravenna

Dipartimento di Matematica 'Tullio Levi-Civita', Università degli Studi di Padova, via Trieste 63,35121 Padova, Italy

Received: August 2016

Revised: October 2016

Published: August 2017

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Abstract

This paper proves new regularity estimates for continuous solutions to the balance equation

${{\partial }_{t}}u+{{\partial }_{x}}f(u)=g\qquad g\ \text{bounded}, f\in {{C}^{\text{2}n}}(\mathbb{R})$

when the flux $f$ satisfies a convexity assumption that we denote as 2n-convexity. The results are known in the case of the quadratic flux by very different arguments in ^[14,10,8]. We prove that the continuity of $u$ must be in fact $1/2n$-Hölder continuity and that the distributional source term $g$ is determined by the classical derivative of $u$ along any characteristics; part of the proof consists in showing that this classical derivative is well defined at any `Lebesgue point' of $g$ for suitable coverings. These two regularity statements fail in general for $C^{\infty}(\mathbb{R})$, strictly convex fluxes, see ^[3].

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Mathematics Subject Classification: Primary: 35L60, 37C10, 58J45.

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Figure 1. Proof of a rough Hölder continuity estimate of u

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Figure 2. Balances on characteristic regions

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