Well-posedness of logarithmic spiral vortex sheets

We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vortr\"age aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for each such spiral the normal component of the velocity field to any spiral remains continuous across the spiral. Moreover, we give a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show 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. Another consequence of the main result is well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane.