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Optimal regularity for all time for entropy solutions of conservation laws in \(BV^s\)

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Abstract

This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional BV spaces. We show that optimality of the regularizing effect for the initial value problem from \(L^\infty \) to fractional Sobolev space and fractional BV spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover, we prove sharp smoothing in \(BV^s\) for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.

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Acknowledgements

Authors thank the anonymous referee for valuable comments and suggestions. The first author would like to thank Inspire faculty-research Grant DST/INSPIRE/04/2016/000237. Authors would like to thank the IFCAM project “Conservation laws: \(BV^s\), interface and control”. The first and third authors acknowledge the support of the Department of Atomic Energy, Government of India, under Project No. 12-R&D-TFR-5.01-0520. The second and the fourth authors would like to thank TIFR-CAM for the hospitality.

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Authors and Affiliations

  1. TIFR, Centre for Applicable Mathematics, Bangalore, India

    Shyam Sundar Ghoshal & Animesh Jana

  2. Université Côte d’Azur, CNRS and Inria, LJAD, Nice, France

    Billel Guelmame & Stéphane Junca

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  1. Shyam Sundar Ghoshal
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  2. Billel Guelmame
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  4. Stéphane Junca
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Correspondence to Shyam Sundar Ghoshal.

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Ghoshal, S.S., Guelmame, B., Jana, A. et al. Optimal regularity for all time for entropy solutions of conservation laws in \(BV^s\). Nonlinear Differ. Equ. Appl. 27, 46 (2020). https://doi.org/10.1007/s00030-020-00649-5

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