A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation
A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation
复制标题
方向性 Lipschitz 扩展引理，及其在连续性方程的唯一性和拉格朗日性中的应用
DOI：
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发表时间：
2018
影响因子：
1.9
通讯作者：
Gianluca Crippa
中科院分区：
文献类型：
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作者：
L. Caravenna;Gianluca Crippa
摘要我们证明了一个Lipschitz扩张引理，其中扩张过程同时保持两个不相等距离的Lipschitz连续性。所考虑的两个距离是欧几里得距离，粗略地说，是沿着连续向量场（可能是多值）流的积分曲线的测地线距离（沿着）。扩张的测地距离的Lipschitz常数可以根据原函数的测地距离的Lipschitz常数来估计。这个Lipschitz延拓引理允许我们在Sobolev空间中的向量场的情况下，在与向量场相关的所谓“向前-向后积分曲线”对于几乎每个起点都是平凡的假设下，去除DiPerna-Lions连续性方程理论中的唯一性所需的解的高可积性假设。更确切地说，这样的向量场，我们证明了唯一性和拉格朗日弱解的连续性方程，只是局部可积。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。
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 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.
具有非凸通量 I 的平衡律的欧拉、拉格朗日和布罗德连续解
DOI：
10.48550/arxiv.1512.04863
发表时间：
2015
期刊：
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影响因子：
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作者：
Alberti G
通讯作者：
Alberti G