Mathematical analysis of vorticity fronts

涡度锋面的数学分析

• 批准号: 2594616
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2594616
• 关键词: Mathematical; analysis; vorticity; fronts

We consider an inviscid and incompressible fluid in a two-dimensional channel, modeled by the incompressible Euler equations. The fluid has two superposed 'layers', each with constant vorticity. As the interface between these layers is unknown, this is a so-called 'free boundary' problem. Our goal is to construct steady solutions to this problem, that is solutions whose interfaces travel at constant speed without change of form. These objects are a middle ground between internal waves, with the same geometry but typically different densities in each layer, and vortex patches, which have the same physical boundary conditions but are typically studied in different geometries. The Euler equations are then expressed as a free boundary problem within a cylindrical domain. The first step is to change coordinates so that the equations becomes a problem in a fixed domain, albeit a much more nonlinear one, which is considered as an abstract equation where a real-analytic nonlinear operator defined on an open subset of a suitable Banach space and depending on real parameters, for instance the speed of the wave. Using bifurcation theory we start with simple solutions which correspond to purely horizontal piecewise-linear shear flows in the physical problem.The first major step is to find nearby solutions which are small-amplitude. We calculate the Frechet derivative of the nonlinear operator evaluated at the simple solutions we found earlier, and study its kernel. This leads to a dispersion relation for the problem, which gives some formal indications as to which sorts of solutions can be expected in various parameter regimes. To actually construct these small-amplitude waves, we will use spatial dynamics techniques. Interpreting the steady problem as an (ill-posed) evolution equation in the horizontal variable x, we will construct a finite-dimensional centre manifold for this evolution equation. This centre manifold will contain all bounded solutions of the problem which are sufficiently 'small'. On the centre manifold the problem will reduce to a finite-dimensional ODE, which will be studied using phase space analysis and dynamical systems techniques.

我们考虑一个无粘和不可压缩的流体在二维通道，不可压缩的欧拉方程建模。流体有两个叠加的“层”，每个层都有恒定的涡度。由于这些层之间的界面是未知的，这是一个所谓的“自由边界”问题。我们的目标是构造这个问题的稳定解，即解的界面以恒定的速度移动而不改变形式。这些物体是内波和涡斑之间的中间地带，内波具有相同的几何形状，但每层的密度通常不同，涡斑具有相同的物理边界条件，但通常在不同的几何形状中进行研究。欧拉方程，然后表示为一个自由边界问题内的一个圆柱形区域。第一步是改变坐标，使方程成为一个问题，在一个固定的域，虽然是一个更加非线性的，这被认为是一个抽象的方程，其中一个真正的解析非线性算子定义在一个适当的Banach空间的一个开放的子集，并取决于真实的参数，例如波的速度。利用分叉理论，我们从对应于纯水平分段线性剪切流的简单解开始，第一个主要步骤是找到附近的小振幅解。我们计算的非线性算子的Frechet导数评估的简单的解决方案，我们发现之前，并研究其内核。这导致了问题的色散关系，它给出了一些正式的指示，哪些种类的解决方案可以预期在不同的参数制度。为了实际构建这些小振幅波，我们将使用空间动力学技术。将定常问题解释为水平变量x中的（不适定的）演化方程，我们将为这个演化方程构造一个有限维中心流形。这个中心流形将包含问题的所有足够小的有界解。在中心流形的问题将减少到一个有限维常微分方程，这将是研究使用相空间分析和动力系统技术。