AbstractIn this work we consider the incompressible Euler equations in three dimensions. We impose a helical symmetry and require that the velocity field must be orthogonal to the tangent lines of the helices. This means that, when looking at the vorticity form, the vortex stretching term vanishes and the three dimensional system can be reduced to a two dimensional system.
We are interested in constructing a solution in the stationary case with vorticity which is concentrated around two points in the plane: (p1,0) and (-p1,0) for some p1>0. We follow the ideas of the 2020 paper by Davila et. al. ('Travelling helices and
the vortex filament conjecture in the incompressible Euler equations') and use Liouville profiles to concentrate the vorticity around these two points. We introduce the parameter ε>0 to denote the core sizes of the concentrated vortices and give an explicit construction of a global approximate solution which has vorticity concentrated around the two points.
We begin by focusing on the regions near each of the points. We give justification as to the choices of various parameters involved in the definition of the Liouville profiles and determine the required relation between the concentration parameter $\eps$ and the distance $2p_1$ between the two points in order to obtain a good approximate solution on the inner regions. We then extend this approximation to outside the inner regions to obtain a global approximate solution.
Finally we comment about how one would expect to be able to use our global approximate solution to construct a true solution to the problem.
|Date of Award||25 May 2022|
|Supervisor||Monica Musso (Supervisor) & Miles Wheeler (Supervisor)|