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# Fractional BV spaces and first applications to scalar conservation laws

Abstract: The aim of this paper is to obtain new fine properties of entropy solutions of nonlinear scalar conservation laws. For this purpose, we study some “fractional BV spaces” denoted $BV^s$, for $0 < s ≤ 1$, introduced by Love and Young in 1937. The $BV^s(\mathbb{R})$ spaces are very closed to the critical Sobolev space $W^{s,1/s}(\mathbb{R})$. We investigate these spaces in relation with one-dimensional scalar conservation laws. $BV^s$ spaces allow to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with $BV^s$ initial data. Furthermore, for the first time we get the maximal $W^{s,p}$ smoothing effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear degenerate convex fluxes.

Paper:
Available as PDF (420 Kbytes).
Author(s):
Christian Bourdarias,
Marguerite Gisclon,
Stephane Junca,
Publishing information:
Published in J. Hyperbolic Differ. Equ. 11, no. 4, p. 655–677, (2014).
Submitted by:
; 2015-03-28.