Continuity equations and ODE flows with non-smooth velocity
Continuity equations and ODE flows with non-smooth velocity
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DOI：
10.1017/s0308210513000085
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发表时间：
2014-12-01
影响因子：
1.3
通讯作者：
Crippa, Gianluca
中科院分区：
文献类型：
--
作者：
Ambrosio, Luigi;Crippa, Gianluca
在这篇文章中，我们回顾了柯西问题、连续性方程和传输方程以及常微分方程(ODE)的适定性理论的许多方面。在这个框架中，我们处理的速度场不是光滑的，但有适当的“弱可微性”假设。我们首先探讨了在非常一般的非光滑环境下偏微分方程与常微分方程组之间的联系。然后，我们讨论了偏微分方程组的重整化性质，并证明了这一性质对Sobolev速度场和有界变差速度场都成立。最后，我们提出了一种基于定量估计的常微分方程组理论的方法。
In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable 'weak differentiability' assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.