Non-minimizing axially symmetric cavity flow

Cavity flow problems in two dimensions, as well as in the axially symmetric three-dimensional case, have been extensively studied in the literature from a qualitative perspective. While numerous results exist concerning minimizers or stable solutions-particularly regarding the regularity of the free boundary and the analysis of singularities—much less is known about the critical points of the corresponding energy functional. In this paper, we focus on investigating the properties of such critical points in the axially symmetric cavity flow problem with a free boundary, in relation to the known variational solutions. Moreover, our approach extends naturally to the case of jet flow problems.