Mathematical analysis of vorticity fronts

我们考虑二维通道中不可压缩的无粘流体，用不可压缩欧拉方程建模。流体有两个重叠的“层”，每一层都有恒定的涡度。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。我们的目标是为这个问题构建稳定的解决方案，即解决方案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旋斑块之间的中间地带，内波具有相同的几何形状，但每一层的密度通常不同，涡旋斑块具有相同的物理边界条件，但通常以不同的几何形状进行研究。然后将欧拉方程表示为圆柱形域内的自由边界问题。第一步是改变坐标，使方程成为一个固定域中的问题，尽管是一个更加非线性的问题，它被认为是一个抽象方程，其中一个实解析非线性算子定义在一个合适的巴拿赫空间的开放子集上，并依赖于实参数，例如波的速度。利用分岔理论，我们从物理问题中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。我们计算了非线性算子的Frechet导数，并研究了它的核。这导致了问题的色散关系，它给出了一些形式的指示，说明在各种参数条件下可以期望哪种类型的解。为了构造这些小振幅的波，我们将使用空间动力学技术。将稳定问题解释为水平变量x中的（不适定）演化方程，我们将为该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ODE，这将研究使用相空间分析和动力系统技术。

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.