Fractional spaces and conservation laws

Description

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 $BV^s$ smoothing effect is more precise and optimal. It implies the optimal Sobolev smoothing effect in $W^{ s,1}$ and also in $W^{ s,p}$ with the optimal $ p = 1/s$. Moreover, the proof expounded does not use the Lax-Oleinik formula but a generalized one-sided Oleinik condition.

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