Mathematics Colloquium
Wojciech Ozanski
University of Southern California
Title: Well-posedness of logarithmic spiral vortex sheets.
Date: Monday, February 28, 2022
Place and Time: Zoom, 3:05pm
Abstract.
We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vortrage aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spirals: a velocity matching condition and a pressure matching condition. Furthermore we discuss that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also discuss well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary, as well as existence of nonsymmetric spirals.