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Optimal regularity for all time for entropy solutions of conservation laws in $BV^s$

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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Submitted on : Thursday, August 13, 2020 - 5:36:58 PM
Last modification on : Friday, August 14, 2020 - 3:33:03 AM

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Shyam Sundar Ghoshal, Billel Guelmame, Animesh Jana, Stéphane Junca. Optimal regularity for all time for entropy solutions of conservation laws in $BV^s$. Nonlinear Differential Equations and Applications, Springer Verlag, 2020, 27 (5), pp.46. ⟨10.1007/s00030-020-00649-5⟩. ⟨hal-02495036v2⟩

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