A numerical study of some questions in vortex rings theory

Critical point approximation through exact regularization

“…Proof. Using the same argument as in the proof of Theorem 2.7, one obtains J*(Pk + i)<f(wk + i) + S*(Pk + i) -f Pk + iwk + idx (5)(6)(7)(8)(9)(10)(11)(12) X r J*{Pk) -2 J \Pk -Pk + \\ dx *k > °-'a Furthermore, J* is coercive and bounded from below in dK. Indeed, for any q g dK, one has (here c and c are positive constants and S is the inverse of -A with homogeneous Dirichlet conditions):…”

“…Proof. Clearly, (2.4) is equivalent to (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) f(uk+i) +f*{B*Pk) = (pk,Buk + l)H, (2.12) Pk+1 g dg(Buk + 1 + X(pk-pk+l)). The last formula can also be written as (2.13) Pk + i = Arg rmnl^g*(q) + ^\q -pk\2H -(q, Buk + l)Hy…”

“…Semilinear elliptic problems of this kind have been studied by many authors (see [3], [5], [6], [8], [16], [28]). In (6.1), the original domain (unbounded) has been approximated by ß, as in [6], [8].…”

“…Again, (6.5) is a formal generalization of the fixed-point method used by H. Berestycki et al in [8]. The main advantages of this procedure are that it permits the numerical treatment of the discontinuous nonlinearity and also that a convergence result has been derived.…”