On the existence of convex classical solutions to a generalized PrandtlBatchelor freeboundary problemII
Abstract
We give an analytical proof of the existence of convex classical solutions for the (convex) PrandtlBatchelor free boundary problem in fluid dynamics. In this problem, a convex vortex core of constant vorticity μ >0 is embedded in a closed irrotational flow inside a closed, convex vessel in ℜ ^{2}. The unknown boundary of the vortex core is a closed curve Γ along which (v^{+)^2(v^)^2=Λ }, where v^{+} and v^{} denote, respectively, the exterior and interior flowspeeds along Γ and Λ is a given constant. Our existence results all apply to the natural multidimensional mathematical generalization of the above problem. The present existence theorems are the only ones available for the PrandtlBatchelor problem for Λ >0, because (a) the author's prior existence treatment was restricted to the case where Λ <0, and because (b) there is no analytical existence theory available for this problem in the nonconvex case, regardless of the sign of Λ .
 Publication:

Zeitschrift Angewandte Mathematik und Physik
 Pub Date:
 2002
 DOI:
 10.1007/s0003300281636
 Bibcode:
 2002ZaMP...53..438A