Weak vorticity formulation for the incompressible Euler equations in domains with boundary
Weak vorticity formulation for the incompressible Euler equations in domains with boundary
复制标题
有边界域中不可压缩欧拉方程的弱涡度公式
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
F. Sueur
中科院分区：
文献类型：
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作者：
D. Iftimie;M. L. Filho;H. N. Lopes;F. Sueur
在本文中，我们研究了具有致密材料边界的不可压缩二维流的相互作用。我们的重点是速度在边界分量周围循环的动态行为，以及流动涡度和边界环流之间可能的交换。我们首先展示了速度可以从涡度和边界分量环流中唯一地重建，这允许使用涡度和环流作为动态变量来重新定义二维欧拉演化。这种涡动力学公式的弱形式称为弱涡量公式。本文的主要结果是弱速度和弱涡量公式之间的等价，没有符号假设。接下来，我们将重点放在通过缓和初始数据并通过极限获得的弱解上，其中涡量相对于勒贝格测度的奇异部分假定为非负。对于这些解，我们证明了每个边界分量周围的环流不能小于初始数据环流，使得非负涡量可能被边界吸收，而不是由边界产生。此外，我们证明了如果弱解在边界分量处保持环流，则它是一个边界耦合弱解，弱涡量公式的一个更强版本。证明了当初始涡度可积时，在边界分量处守恒环流的弱解的存在性。此外，我们还讨论了流体对材料边界构件施加的机械力的定义及其与循环守恒的关系。最后，我们描述了一个有界有洞区域的相应结果，以及证明中需要的适应性。
In this article we examine the interaction of incompressible 2D flows with compact 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 We begin by showing 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. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Next, we focus on 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 that the circulations around each boundary component cannot be smaller than the initial data circulation, so that nonnegative vorticity may be absorbed by the boundary, but not produced by the boundary. In addition, we prove that if the weak solution conserves circulation at the boundary components 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 at the boundary components if the initial vorticity is integrable. In addition, we discuss the definition of the mechanical force which the flow exerts on material boundary components and its relation with conservation of circulation. Finally, we describe the corresponding results for a bounded domain with holes, and the adaptations required in the proofs.