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On optimal regularity estimates for finite-entropy solutions of scalar conservation laws
Xavier Lamy1  ; Andrew Lorent2 ; Guanying Peng3
1 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPSIMT, F-31062 Toulouse Cedex 9, France.
2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA.
3 Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 599-608.

We consider finite-entropy solutions of scalar conservation laws u t +a(u) x =0, that is, bounded weak solutions whose entropy productions are locally finite Radon measures. Under the assumptions that the flux function a is strictly convex (with possibly degenerate convexity) and a ′′ forms a doubling measure, we obtain a characterization of finite-entropy solutions in terms of an optimal regularity estimate involving a cost function first used by Golse and Perthame.

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DOI : 10.5802/crmath.427
Classification : 35L65
Xavier Lamy 1 ; Andrew Lorent 2 ; Guanying Peng 3

1 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPSIMT, F-31062 Toulouse Cedex 9, France.
2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA.
3 Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Xavier Lamy; Andrew Lorent; Guanying Peng. On optimal regularity estimates for finite-entropy solutions of scalar conservation laws. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 599-608. doi : 10.5802/crmath.427. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.427/

[1] Luigi Ambrosio; Nicola Fusco; Diego Pallara Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, 2000

[2] Claude Bardos; Piotr Gwiazda; Agnieszka Świerczewska-Gwiazda; Edriss S. Titi; Emil Wiedemann On the extension of Onsager’s conjecture for general conservation laws, J. Nonlinear Sci., Volume 29 (2019) no. 2, pp. 501-510 | DOI | MR | Zbl

[3] Giovanni Bellettini; Lorenzo Bertini; Mauro Mariani; Matteo Novaga Γ-entropy cost for scalar conservation laws, Arch. Ration. Mech. Anal., Volume 195 (2010) no. 1, pp. 261-309 | DOI | MR | Zbl

[4] Camillo De Lellis; Radu Ignat A regularizing property of the 2D-eikonal equation, Commun. Partial Differ. Equations, Volume 40 (2015) no. 8, pp. 1543-1557 | DOI | MR | Zbl

[5] Camillo De Lellis; Michael Westdickenberg On the optimality of velocity averaging lemmas, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 20 (2003) no. 6, pp. 1075-1085 | Numdam | MR | Zbl

[6] Francesco Ghiraldin; Xavier Lamy Optimal Besov differentiability for entropy solutions of the eikonal equation, Commun. Pure Appl. Math., Volume 73 (2020) no. 2, pp. 317-349 | DOI | MR | Zbl

[7] François Golse; Benoît Perthame Optimal regularizing effect for scalar conservation laws, Rev. Mat. Iberoam., Volume 29 (2013) no. 4, pp. 1477-1504 | DOI | MR | Zbl

[8] Stanislas N. Kružkov First order quasilinear equations with several independent variables, Mat. Sb., N. Ser., Volume 81 (123) (1970), pp. 228-255

[9] Xavier Lamy; Andrew Lorent; Guanying Peng On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds, Commun. Partial Differ. Equations, Volume 47 (2022) no. 11, pp. 2270-2309 | DOI | MR | Zbl

[10] Andrew Lorent; Guanying Peng Factorization for entropy production of the Eikonal equation and regularity (2021) (https://arxiv.org/abs/2104.01467)

[11] Elio Marconi On the structure of weak solutions to scalar conservation laws with finite entropy production, Calc. Var. Partial Differ. Equ., Volume 61 (2022) no. 1, p. 30 | MR | Zbl

[12] Elio Marconi The rectifiability of the entropy defect measure for Burgers equation, J. Funct. Anal., Volume 283 (2022) no. 6, p. 109568 | DOI | MR | Zbl

[13] Mauro Mariani Large deviations principles for stochastic scalar conservation laws, Probab. Theory Relat. Fields, Volume 147 (2010) no. 3-4, pp. 607-648 | DOI | MR | Zbl

[14] Srinivasa R. S. Varadhan Large deviations for the asymmetric simple exclusion process, Stochastic analysis on large scale interacting systems (Advanced Studies in Pure Mathematics), Volume 39, Mathematical Society of Japan, 2004, pp. 1-27 | MR | Zbl

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