Rigidity aspects of singular patches in stratified flows

IF 0.8 Q2 MATHEMATICS Tunisian Journal of Mathematics Pub Date : 2021-11-14 DOI:10.2140/tunis.2022.4.465
T. Hmidi, Haroune Houamed, M. Zerguine
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引用次数: 3

Abstract

We explore the local well-posedness theory for the 2d inviscid Boussinesq system when the vorticity is given by a singular patch. We give a significant improvement of \cite{Hassainia-Hmidi} by replacing their compatibility assumption on the density with a constraint on its platitude degree on the singular set. The second main contribution focuses on the same issue for the partial viscous Boussinesq system. We establish a uniform LWP theory with respect to the vanishing conductivity. This issue is much more delicate than the inviscid case and one should carefully deal with various difficulties related to the diffusion effects which tend to alter some local structures. The weak a priori estimates are not trivial and refined analysis on transport-diffusion equation subject to a logarithmic singular potential is required. Another difficulty stems from some commutators arising in the control of the co-normal regularity that we counterbalance in part by the maximal smoothing effects of transport-diffusion equation advected by a velocity field which scales slightly below the Lipschitz class.
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分层流动中奇异斑块的刚性方面
研究了二维无粘Boussinesq系统在涡度由奇异块给出时的局部适定性理论。我们将\cite{Hassainia-Hmidi}在密度上的相容性假设替换为其在奇异集上的重复度约束,从而对其进行了显著改进。第二个主要贡献集中在部分粘性Boussinesq系统的相同问题上。我们建立了关于电导率消失的统一LWP理论。这个问题比不粘的情况要微妙得多,人们应该仔细处理与扩散效应有关的各种困难,扩散效应往往会改变一些局部结构。弱先验估计不是微不足道的,需要对对数奇异势作用下的输运-扩散方程进行精细化分析。另一个困难来自于在控制共正态规则时产生的一些换向子,我们通过一个尺度略低于Lipschitz类的速度场平流的输运-扩散方程的最大平滑效应来部分地抵消这些换向子。
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来源期刊
Tunisian Journal of Mathematics
Tunisian Journal of Mathematics Mathematics-Mathematics (all)
CiteScore
1.70
自引率
0.00%
发文量
12
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