Flatness-implies-regularity for Neumann-type Bernoulli problems

Establish a flatness-implies-regularity theorem for the Neumann-type Bernoulli free boundary problem governing three-dimensional axially symmetric cavity flows.

Background

The paper emphasizes that Neumann-type Bernoulli problems are substantially less developed than Dirichlet-type Bernoulli problems. In particular, even in the axially symmetric cavity setting, the authors state that well-posedness in the form of a flatness-implies-regularity result has not been established. Such a theorem would provide a fundamental regularity principle for this free boundary problem.

References

In fact, while Dirichlet-type Bernoulli problems, in which the free surface is a level set of the solution, have been extensively researched in the free boundary community, Neumann-type Bernoulli problems are notoriously hard, and not even well-posedness in the sense of a flatness-implies-regularity result is known for this important problem in mathematical physics.

— Complete Rigidity at infinity and Existence of the Levinson Cavity  (2608.18913 - Li et al., 19 Aug 2026) in Section 1, subsection “Cavity flows”