Convergence of the point vortex method for 2-D vortex sheet

Abstract

Abstract. We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

Vortex Liquids and the Ginzburg–landau Equation

“…In the latter case, it was shown by Schochet [44], and later by Liu and Xin [34], that the vortex density function for Euler point vortices on R 2 , which follow the Kirchhoff laẇ The present work is the first to directly couple the Ginzburg-Landau equation to a mean field PDE. All previous works either prove a PDE to ODE limit for a finite number of vortices or pass from the ODE to the mean field PDE limit.…”

“…The proof requires a careful expansion of the time-dependent behavior of 1 n n j=1 δ a j (t) integrated against a test function with compact support. Implementing the strategy of [34,32] and using estimates on the Neumann function, we prove the convergence and the local velocity bound.…”

“…As in [35,34,32], we prove a decay estimate on M r (ω n ) below in order to pass to the limit in the main term A n . 1 + |log r | n M r (ω n ),…”

“…Using 2π δ a k = ∆N n (·, a k ) and the above identity for ∇ S j n yields 1 [34], we define the matrix-valued function K(n, y, z; η):…”

“…We now examine the convergence behavior of A j n and B n . From Lemma 24, one can follow the arguments of [44,34] to establish the convergence of A j n . Looking at A 1 n , and taking χ ∈ C ∞ 0 (Ω) and setting η = ∂ 2 x 1 − ∂ 2 x 2 χ, we have and by Lemma 25 the term goes to zero as n → ∞ and r → 0.…”