Mathematical analysis of 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.