Bachelor's Degree Project Presentation: On the Existence of a Periodic Solution to Point and Patch Setting for the Incompressible 2D Euler Equation

Abstract: We consider the incompressible Euler equations in two dimensions. These are a set of partial differential equations modeling certain ideal fluids. The equations may be formulated in terms of velocity flow or in terms of their curl, which in the context of fluid dynamics is aptly referred to as vorticity. We exhibit a certain simple solution to the equations consisting of a time-invariant circular patch of constant vorticity and remark that points outside rotate around said patch. Motivated by this we try to see whether a periodic point-and-patch solution also exists in the case when there is a localized non-zero vorticity in the point. The problem is reformulated into one of fixed point nature and thus suitable to attack by the Banach fixed point theorem. Some mayor steps towards applying said theorem are taken. Lastly some numerical simulations are performed to try to back up the theoretical results.

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