Abstract

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space $W^{1,p},$ where p is larger than the space dimension, under the assumption that the so-called “forward-backward integral curves” associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable. Additionally, for such vector fields it is possible to prove almost everywhere uniqueness of (standard) integral curves, which also implies uniqueness of positive measure solutions to the continuity equation with absolutely continuous initial datum.

摘要

我们证明了一个 Lipschitz 扩展引理，其中扩展过程同时保持了两个非等价距离的 Lipschitz 连续性。所考虑的两个距离是欧几里得距离，粗略地说，是沿连续矢量场（可能是多值）流的积分曲线的测地线距离。可以根据原始函数的测地线距离的 Lipschitz 常数来估计扩展的测地线距离的 Lipschitz 常数。这个 Lipschitz 扩展引理允许我们在 Sobolev 空间中的向量场的情况下去除连续性方程的 DiPerna-Lions 理论中唯一性所需的解的高可积性假设${宽}^{1,磷},$其中p大于空间维度，假设与向量场相关的所谓“前后积分曲线”对于几乎每个起点都是微不足道的。更准确地说，对于这样的向量场，我们证明了只是局部可积的连续性方程的弱解的唯一性和拉格朗日性。此外，对于这样的向量场，可以证明（标准）积分曲线的几乎处处唯一性，这也意味着具有绝对连续初始数据的连续性方程的正测度解的唯一性。