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

Pierre Castelli (1) , Stéphane Junca (2, 3)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 COFFEE - COmplex Flows For Energy and Environment
3 EDP

Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect for entropy solutions of nonlinear scalar conservations laws . In this short paper we will restrict our attention to the simpler one-dimensional case. First, supercritical geometric optics lead to sequences of $C^\infty$ 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.

Mots clés: scalar conservation laws; geometric optics; regularity; fractional BV spaces; Sobolev spaces

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Castelli, Stéphane Junca. Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws. 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Fabio Ancona, Alberto Bressan, Pierangelo Marcati, Andrea Marson, Jun 2012, Padoue, Italy. pp.709-716. ⟨hal-00785529⟩