CASTELLI , Pierre

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 $W^{ s,1}$ , and in fractional $BV$ spaces: $BV^s$. The...

In 1994, Lions, Perthame and Tadmor conjectured an optimal smoothing effect for entropy solutions of multidimensional scalar conservation laws. This effect estimated in fractional Sobolev spaces is linked to the flux nonlinearity. In order to show that the conjectured smoothing effect cannot be exceeded, we use a new definition of a nonlinear...

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...

This paper deals with a sharp smoothing effect for entropy solutions of one-dimensional scalar conservation laws with a degenerate convex flux. We briefly explain why degenerate fluxes are related with the optimal smoothing effect conjectured by Lions, Perthame, Tadmor for entropy solutions of multidimensional conservation laws. It turns out...

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