Nonlinear desingularization in certain free-boundary problems

Abstract

Abstract. We consider a nonlinear, elliptic, free-boundary problem involving an initially unknown set A that represents, for example, the cross-section of a steady vortex ring or of a confined plasma in equilibrium. The solutions are characterized by a variational principle which allows us to describe their behaviour under a limiting process such that the diameter of A tends to zero, while the solutions degenerate to the solution of a related linear problem. This limiting solution is the sum of the Green function of the linear operator and of a smooth function satisfying the boundary conditions. Mathematically speaking, this limiting process, that we call "nonlinear desingularization", is a novel kind of bifurcation phenomenon since the nonlinear effect here involves smoothing the singularity of the associated linear problem. "Nonlinear desingularization" is an interesting, hitherto little studied phenomenon in nonlinear partial differential equations. By nonlinear desingularization we mean that a linear boundary-value problem, whose solution possesses one or more isolated singularities, is a degenerate form of a family of nonlinear problems whose solutions are smooth and, moreover, converge to the solution of the linear system upon degeneration. This phenomenon occurs, for example, in the study of vortex motion in ideal fluids; there a circular vortex filament is used to approximate a steady vortex ring of small cross-section. The Stokes stream function of the filament is the Green function of an elliptic operator, and hence has an isolated singularity, while the Stokes stream function for the vortex ring of small crosssection satisfies a non-linear partial differential equation, and is smooth. In this paper we study this situation from a global point of view that is independent of the elaborate singular-perturbation techniques often used for such problems.Nonlinear desingularization arises in problems of theoretical physics other than the classical problem mentioned above. For example, it occurs in the onset of flux penetration, as determined by the Ginzberg-Landau theory, in supercon- has discussed it in connection with new classes of static, Euclidean, SU(2) Yang-Mills fields (monopoles) that are not spherically synmaetric and have unit Pontrjagin index. We hope to consider these applications in a later paper.

A numerical study of some questions in vortex rings theory

“…Therefore, it follows from (3.6) that For the solutions of vortex ring problems some intégral identities can be derived which will further prövide the possibility of useful checks ~ön the numerical work (see Section 10 below). These identities were first reported in M. S. Berger and L. E. Fraenkel [15] and A. Friedman and B. Turkington [29] in the three-dimensional case. For simplicity, we will only refer to the twodimensional problem.…”

Multi-peak solutions to two types of free boundary problems

“…In [1,2,27,38], the solutions were obtained by using mountain pass lemma for various nonlinearities f (x, u) and any ε > 0. In [3,6,17,28,35], to find the solutions, the constrained variation methods were used, but the vorticity function f is unknown a priori. Moreover, in [6,17,28], the solutions were obtained by regarding 1/ε 2 as eigenvalue, so ε is not arbitrary.…”

“…In [3,6,17,28,35], to find the solutions, the constrained variation methods were used, but the vorticity function f is unknown a priori. Moreover, in [6,17,28], the solutions were obtained by regarding 1/ε 2 as eigenvalue, so ε is not arbitrary. The asymptotic behavior of the solution pair (u γ , A γ ) of problem (1.6) was investigated in [6,8,12,24,38].…”

“…Moreover, in [6,17,28], the solutions were obtained by regarding 1/ε 2 as eigenvalue, so ε is not arbitrary. The asymptotic behavior of the solution pair (u γ , A γ ) of problem (1.6) was investigated in [6,8,12,24,38]. Recently, for the case f (t) = t p−1 and N ≥ 2, under the condition that…”

Equilibrium vortex configurations in domains with boundary

“…It was shown by Berger and Fraenkel [1] and Turkington [10] that if the vorticity is sufficiently concentrated, then the flow is a desingularization of a point vortex flow. Conversely, any configuration of point vortices in stable equilibrium can be desingularized ([11], [12], [3])-Nevertheless, configurations of point vortices in equilibrium with respect to a fixed boundary have not been studied extensively.…”