Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two

Abstract

Abstract. In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem

Planar vortex patch problem in incompressible steady flow

“…The involvement of the measure implies that the estimates needed in the proof of Theorem 1.1 are domain-variation type estimates. These estimates are entirely different from those estimates in [11,12,15], which are all carried out in some standard Sobolev spaces.…”

“…To prove Theorem 1.1, although we use a finite reduction argument as in [11,12,15], we need to deal with some serious difficulties when the nonlinear term is discontinuous, which is more physically appropriate than the problem studied in [11,27]. Mathematically, the nonlinear term 1 u>κ may be regarded as…”

“…For the earlier work, we refer to the remarkable papers of Fraenkel and Berger [18] and Norbury [25,26]. The advantage of a finite dimensional reduction argument, once it can be used, is that solutions concentrating at a saddle points, or concentrating at several points can be constructed [11,12,15]. But if f (y, u) is not C 1 , it is very delicate to use the finite dimensional reduction argument even in the case that f (y, u) is continuous.…”

“…See for example [12] in the construction of solutions for the plasm problem. If f (y, u) is C 1 and convex, the readers can refer to the results in [11,15,16,20,21,23,24,27,30] and the references therein.…”

Desingularization of Vortex Rings and Shallow Water Vortices by a Semilinear Elliptic Problem

“…Physically, this takes into account that the total flow between the two vortices should blow up as the logarithm of the diameter of the vortex core. They have obtained a desingularization result for solutions constructed by variational methods; solutions to the same problem where also obtained by Lyapunov-Schmidt reduction argument [18,19].…”