论文标题
躺在非平滑矢量领域的括号和流动的流向
Lie brackets of nonsmooth vector fields and commutation of their flows
论文作者
论文摘要
众所周知，如果这些矢量场的谎言括号消失，则由两个平滑的向量场产生的流动产生的流量。众所周知，这种断言会扩展到Lipschitz的连续矢量场，直到在几乎所有地方平等的意义上解释其谎言括号的消失。我们表明，这不能扩展到将军A.E.承认A.E.的可区分矢量领域独特的流。但是，我们表明，当一个字段是Lipschitz连续的，而另一个仅是Sobolev常规（但承认常规的Lagrangian流动）时，该扩展就成立了。
It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this cannot be extended to general a.e. differentiable vector fields admitting a.e. unique flows. We show however that the extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (but admitting a regular Lagrangian flow).
