PLANAR VORTICES IN A BOUNDED DOMAIN WITH A HOLE

Authors：Yan, Shusen^*; Yu, Weilin

Source：Electronic Research Archive, 2021, 29(6): 4229-4241.

DOI：10.3934/era.2021081

• The Euler flow
• semilinear elliptic equation
• variational method
• free boundary problem
• reduction

In this paper, we consider the inviscid, incompressible planar flows in a bounded domain with a hole and construct stationary classical solutions with single vortex core, which is closed to the hole. This is carried out by constructing solutions to the following semilinear elliptic problem @@@ { -Delta psi = lambda(psi - kappa/4 pi ln lambda)(+)(p), in Omega, @@@ psi = rho(lambda), on partial derivative O-0, (1) @@@ psi = 0, on partial derivative Omega(0), @@@ where p > 1, kappa is a positive constant, rho(lambda) is a constant, depending on lambda, Omega = Omega(0) \ O over line 0 and Omega(0), O-0 are two planar bounded simply-connected domains. We show that under the assumption (ln lambda)Sigma < rho(lambda) < (ln lambda)(1-sigma) for some sigma > 0 small, (1) has a solution psi(lambda), whose vorticity set {y is an element of Omega : psi(y)-kappa+rho(lambda)eta(y) > 0} shrinks to the boundary of the hole as lambda -> +infinity.

• Institution
  中国科学院