论文标题
电流和几何rademacher型定理的运输
Transport of currents and geometric Rademacher-type theorems
论文作者
论文摘要
可以用几何（Lie）传输方程来表达多种媒介中多种奇异结构的运输，例如流体中的涡流点/线/表格，晶体塑料中的脱位环或磁性的拓扑奇异性。 \ frac {\ mathrm {d}} {\ mathrm {d} t} t} t_t + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {l} _ {b_t} t_t = 0 \] t_t = 0 \]，用于一个零用或正常$ k $ k $ k $ -currents $ c $ t \ mapsto $ t \ t_t_t $ t_t $ t_ $ t_ $ n $ n $ n $ n $ k $ n $ n $ n $ n。在这里，$ b_t $是驱动矢量字段，$ \ mathcal {l} _ {b_t} t_t $是$ t_t $相对于$ b_t $的lie导数。该PDE以不同值的坐标编写，该PDE涵盖了经典的传输方程（$ k = D $），连续性方程（$ k = 0 $），以及用于晶体中位错线（$ k = 1 $）和液体中的膜中的脱位线的方程式（$ k = d-1 $）。对于常规拉格朗日流动的二型二角河和安布罗西奥理论，高层和底层的案例引起了很多关注。另一方面，目前在中等差异的情况下，几乎没有严格知道。这项工作开发了用于任意$ k $的几何传输方程的理论，在无边界电流的情况下，$ t_t $，尤其涵盖了解决方案，结构定理，重新讨论性以及许多rademacher型不同的可不同性结果的独特性。后者的产量，给定绝对连续的（时间）路径$ t \ mapsto t_t $，几乎到处都是“几何导数”的存在，即驱动向量场$ b_t $。这个微妙的问题证明，与进化的关键集相关，这是这项工作中引入的一个新概念，这与Sard的定理密切相关，并且关注“及时涂抹”的奇异性。我们的不同性结果是尖锐的，我们通过明确的例子证明了这一点。
The transport of many kinds of singular structures in a medium, such as vortex points/lines/sheets in fluids, dislocation loops in crystalline plastic solids, or topological singularities in magnetism, can be expressed in terms of the geometric (Lie) transport equation \[ \frac{\mathrm{d}}{\mathrm{d} t} T_t + \mathcal{L}_{b_t} T_t = 0 \] for a time-indexed family of integral or normal $k$-currents $t \mapsto T_t$ in $\mathbb{R}^d$. Here, $b_t$ is the driving vector field and $\mathcal{L}_{b_t} T_t$ is the Lie derivative of $T_t$ with respect to $b_t$. Written in coordinates for different values of $k$, this PDE encompasses the classical transport equation ($k = d$), the continuity equation ($k = 0$), as well as the equations for the transport of dislocation lines in crystals ($k = 1$) and membranes in liquids ($k =d-1$). The top-dimensional and bottom-dimensional cases have received a great deal of attention in connection with the DiPerna--Lions and Ambrosio theories of Regular Lagrangian Flows. On the other hand, very little is rigorously known at present in the intermediate-dimensional cases. This work develops the theory of the geometric transport equation for arbitrary $k$ and in the case of boundaryless currents $T_t$, covering in particular existence and uniqueness of solutions, structure theorems, rectifiability, and a number of Rademacher-type differentiability results. The latter yield, given an absolutely continuous (in time) path $t \mapsto T_t$, the existence almost everywhere of a ''geometric derivative'', namely a driving vector field $b_t$. This subtle question turns out to be intimately related to the critical set of the evolution, a new notion introduced in this work, which is closely related to Sard's theorem and concerns singularities that are ''smeared out in time''. Our differentiability results are sharp, which we demonstrate through an explicit example.
