一维非线性标量守恒律的振荡波和最佳光滑效应
Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws
摘要：
Lions，Perthame，Tadmor在1994年证明了非线性标量守恒律熵解的最优光滑效应.在这篇短文中，我们将把注意力限制在较简单的一维情形。首先，超临界几何光学导致序列的$C^\infty$的解决方案一致有界的索伯列夫空间。其次给出了恰属于合适的索伯列夫空间的连续解。为此，我们给出了两个新的非线性通量的定义，并引入了分数$BV$空间。
英文摘要：
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.
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