The paper proves new regularity estimates for continuous solutions to balance equation u_t+f(u)_x=g where the source g is bounded and the flux f is 2n-times continuously differentiable and it 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 the literature. We prove that the continuity of u must be in fact 1/2n-Holder 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 strictly convex fuxes, see [3].
Regularity estimates for continuous solutions of alpha-convex balance laws
CARAVENNA, LAURA
2017
Abstract
The paper proves new regularity estimates for continuous solutions to balance equation u_t+f(u)_x=g where the source g is bounded and the flux f is 2n-times continuously differentiable and it 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 the literature. We prove that the continuity of u must be in fact 1/2n-Holder 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 strictly convex fuxes, see [3].Pubblicazioni consigliate
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