Fractional BV spaces and first applications to scalar conservation laws
Fractional BV spaces and first applications to scalar conservation laws
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分数 BV 空间和标量守恒定律的首次应用
DOI：
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发表时间：
2013
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通讯作者：
S. Junca
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作者：
C. Bourdarias;M. Gisclon;S. Junca
本文的目的是得到非线性标量守恒律熵解的新的精细性质。为了这个目的，我们研究了Love和Young在1937年提出的一些“分数BV空间”，记为BV^s$，其中0 < s leq 1$。BV^s（mathbb{R}）空间非常接近临界Sobolev空间W^{s，1/s}（mathbb{R}）。我们调查这些空间与一维标量守恒律。$BV^s$ spaces允许使用比BV函数更不规则的函数，并且在这种情况下看起来更自然。我们得到了具有$BV^s$初始数据的熵解的稳定性结果。此外，我们还首次得到了由P. L.狮子，B。Perthame和E.所有非线性退化凸通量的Tadmor。
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 $0 < s leq 1$, 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.