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Preprint 2015-013

Oscillating waves and optimal smoothing effect for one- dimensional nonlinear scalar conservation laws

Pierre Castelli and Stephane Junca

Abstract: Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect for entropy solutions of nonlinear scalar conservations laws ([19]). In this short paper we will restrict our attention to the simpler one-dimensional case. First, supercritical geometric optics lead to sequences of C solutions uniformly bounded in the Sobolev space conjectured. Second we give continuous solutions which belong exactly to the suitable Sobolev space. In order to do so we give two new definitions of nonlinear flux and we introduce fractional BV spaces.

Reference
[19] P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7, (1994), 169–192 [MR1201239].
Paper:
Available as PDF (264 Kbytes).
Author(s):
Pierre Castelli,
Stephane Junca,
Publishing information:
Published in AIMS Ser. on Appl. Math. 8, 709–716, (2014).
Submitted by:
; 2015-04-16.