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

Abstract

Abstract. Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on a study of solutions to the semilinear elliptic problemfor small values of ε > 0.

Multiple solutions for a free boundary problem arising in plasma physics

“…Moreover, the plasma ring is very similar to the vortex ring in an ideal fluid (for details, we refer the reader to [1,3,10,12,13,20,21] and the references therein). This equation has its origin in plasma physics and describes the equilibrium of a plasma confined in a tokamak.…”

“…The plasma ring can be approximated by a cylinder. Moreover, the plasma ring is very similar to the vortex ring in an ideal fluid (for details, we refer the reader to [1,3,10,12,13,20,21] and the references therein).…”

Vortex Motion for the Lake Equations

“…In the stationary case for the lake equations, where the velocity u does not depend on the time t (1.1), there exist families of stationary solutions concentrated at a point of maximal depth or at a point where the irrotational flow generated by a boundary condition of order ln 1 ε balances the diverging motion of (1.3) [13][14][15]. (Corresponding results were already known for the planar Euler equations [3,7,[48][49][50].)…”

Desingularization of steady vortex of perturbation type in the lake equations

“…(1. 13) In order to solve this system, as in [14], we assume q ∈ C 1 ( D) ∩ C 2 (D) be the solution of…”

“…For the lake model, there are less related work. Using the stream-function method, De Valeriola-Van Schaftingen [14] constructed a family of desingularized solutions of equation (1.13) when f (s) = s p + for some p > 1 and δ(ε) ≈ 1. Dekeyser [12] [13] also investigated the problem by the vorticity method.…”