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March  2017, 16(2): 629-644. doi: 10.3934/cpaa.2017031

## Regularity estimates for continuous solutions of α-convex balance laws

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

Received  August 2016 Revised  October 2016 Published  January 2017

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].
Citation: Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure & Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031
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##### References:
Proof of a rough Hölder continuity estimate of u
Balances on characteristic regions
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