TY - GEN

T1 - Fractional spaces and conservation laws

AU - Castelli, Pierre

AU - Jabin, Pierre Emmanuel

AU - Junca, Stéphane

N1 - Funding Information:
We thank the support of the project SlowDyn (team leader Bruno Lombard): an interdisciplinary CNRS project based on the LMA (CNRS, UPR 7051, Marseille). P-E Jabin acknowledges the support of NSF Grants DMS 1312142 and 1614537 and by NSF Grant RNMS (Ki-Net) 1107444.
Funding Information:
Acknowledgements We thank the support of the project SlowDyn (team leader Bruno Lombard): an interdisciplinary CNRS project based on the LMA (CNRS, UPR 7051, Marseille). P-E Jabin acknowledges the support of NSF Grants DMS 1312142 and 1614537 and by NSF Grant RNMS (Ki-Net) 1107444.
Publisher Copyright:
© Springer International Publishing AG, part of Springer Nature 2018.

PY - 2018

Y1 - 2018

N2 - In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar conservation laws in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case, we detail the proof of this conjecture in the framework of Sobolev fractional spaces Ws,1, and in fractional BV spaces: BVs. The BVs smoothing effect is more precise and optimal. It implies the optimal Sobolev smoothing effect in Ws,1 and also in Ws,p with the optimal p=1/s. Moreover, the proof expounded does not use the Lax–Oleinik formula but a generalized one-sided Oleinik condition.

AB - In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar conservation laws in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case, we detail the proof of this conjecture in the framework of Sobolev fractional spaces Ws,1, and in fractional BV spaces: BVs. The BVs smoothing effect is more precise and optimal. It implies the optimal Sobolev smoothing effect in Ws,1 and also in Ws,p with the optimal p=1/s. Moreover, the proof expounded does not use the Lax–Oleinik formula but a generalized one-sided Oleinik condition.

UR - http://www.scopus.com/inward/record.url?scp=85049372807&partnerID=8YFLogxK

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U2 - 10.1007/978-3-319-91545-6_23

DO - 10.1007/978-3-319-91545-6_23

M3 - Conference contribution

AN - SCOPUS:85049372807

SN - 9783319915449

T3 - Springer Proceedings in Mathematics and Statistics

SP - 285

EP - 293

BT - Theory, Numerics and Applications of Hyperbolic Problems I - Aachen, Germany, 2016

A2 - Westdickenberg, Michael

A2 - Klingenberg, Christian

PB - Springer New York LLC

T2 - 16th International Conference on Hyperbolic Problems: Theory, Numerics and Applications, 2016

Y2 - 1 August 2016 through 5 August 2016

ER -