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We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.
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Xavier Lamy, Benjamin Gess, Regularity of solutions to scalar conservation laws with a force. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, pp. 505–521DOI 10.1016/J.ANIHPC.2018.07.002