Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains
Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains
复制标题
有界域中不可压缩二维欧拉方程的弱涡度公式
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
H. N. Lopes
中科院分区：
文献类型：
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作者：
D. Iftimie;M. L. Filho;H. N. Lopes
摘要在这篇文章中，我们研究了不可压缩二维流动与材料边界的相互作用。我们的重点是在有界区域的情况下，边界组件周围的速度循环的动态行为和流动涡量和边界环流之间的可能的交换与涡面初始数据的流动。我们的出发点是观察到，理想的流动与涡面规则性有明确的环流周围的每个连接的组成部分的边界。此外，我们表明，速度可以唯一重建的涡度和边界分量的环流，这使得重铸2D欧拉演化使用涡度和环流作为动态变量。这种涡动力学方程的弱形式称为弱涡度方程。当初始涡度是满足符号条件的有界测度时，德洛尔定理保证了速度形式的二维欧拉方程弱解的存在性。本文的主要结果是，在没有符号假设的情况下，弱速度和弱涡量公式是等价的。尽管他们是等价的，定性信息弱解是更明显的弱涡制定比从速度制定，和文章的其余部分是专门的几个后果，可以从我们的主要结果。首先，我们考虑通过软化初始数据并传递到极限而获得的弱解，其中涡度奇异部分相对于勒贝格测度假设为非负。对于这些解决方案，我们证明了一组不等式，限制可能产生的涡的边界。其次，我们证明了，如果弱解保持周围的边界分量的环流，那么它是一个边界耦合弱解，弱涡公式的一个更强的版本。我们证明存在一个弱的解决方案，保持周围的边界组件的流通，如果初始涡是可积的，即如果奇异部分消失。最后讨论了流体作用于各物质边界元的净机械力的定义及其与环流守恒的关系。
Abstract In this article we examine the interaction of incompressible 2D flows with material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data, in the case of bounded domains. Our point of departure is the observation that ideal flows with vortex sheet regularity have well-defined circulation around each connected component of the boundary. In addition, we show that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. Existence of a weak solution for the 2D Euler equations, in velocity form, is guaranteed by Delort’s Theorem, when the initial vorticity is a bounded measure satisfying a sign condition. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Despite their being equivalent, the qualitative information concerning weak solutions is more apparent from the weak vorticity formulation than from the velocity formulation, and the remainder of the article is devoted to several consequences which can be derived from our main result. First, we consider weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove a set of inequalities which restrict the possible generation of vorticity by the boundary. Next, we prove that, if the weak solution conserves circulation around the boundary components, then it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation around the boundary components if the initial vorticity is integrable, i.e. if the singular part vanishes. Finally, we discuss the definition of the net mechanical force which the flow exerts on each material boundary component and its relation with conservation of circulation.