Growth of Sobolev norms and loss of regularity in transport equations
Growth of Sobolev norms and loss of regularity in transport equations
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DOI：
10.1098/rsta.2021.0024
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发表时间：
2022-06-13
期刊：
影响因子：
5
通讯作者：
Mazzucato, Anna L.
中科院分区：
文献类型：
--
作者：
Crippa, Gianluca;Elgindi, Tarek;Mazzucato, Anna L.
我们考虑由不规则的无发散矢量场平流的被动标量的运输。给定任意非常数的初始数据是H-loc（1）（R-d）的元素，d >= 2，我们构造了一个无发散的平流速度场v（依赖于），使得输运方程的唯一弱解在任何正时间都不属于H-loc（1）（R-d）。速度场v是光滑的，除了在一个点，在时间上均匀控制，并且属于几乎所有不嵌入Lipschitz类的Sobolev空间W-s，W-p。速度场v是通过拉回和重新缩放环面上取决于初始数据的正弦/余弦剪切流序列来构造的。这种规律性损失的结果补充了Ann. PDE，5（1）：2019年9月19日论文中的结果。这篇文章是“物理流体动力学中的数学问题（第1部分）”主题的一部分。
We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data is an element of H-loc(1)(R-d), d >= 2, we construct a divergence-free advecting velocity field v (depending on ) for which the unique weak solution to the transport equation does not belong to H-loc(1)(R-d) for any positive time. The velocity field v is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space W-s,W-p that does not embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE, 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.