Levinson cavity asymptotics problem

For the three-dimensional axially symmetric steady incompressible cavity-flow free-boundary problem, determine whether the cavity solutions whose existence was established by Garabedian–Lewy–Schiffer have free boundaries with Levinson asymptotic profile $(\log r)^{-1/4}\sqrt{r}$ as $r\to\infty$ ; at minimum, determine whether there exists at least one such cavity solution having this asymptotic behavior.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Complete Rigidity at infinity and Existence of the Levinson Cavity — arXiv

1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

An unrefereed preprint claims to resolve Levinson’s 80-year-old cavity-asymptotics question, but the result has not been independently verified.

The problem concerns the free-boundary asymptotics left open under Levinson’s restrictive hypotheses. The new claim asserts convergence to the profile $\sqrt{r}(\log r)^{-1/4}$ after reducing the problem to an ordinary differential equation.

August 2026 claimed resolution

The preprint claims complete rigidity at infinity and existence of the Levinson cavity, proving convergence to $\sqrt{r}(\log r)^{-1/4}$ . It is an unrefereed preprint, and no independent verification or published corroboration was found in the retrieved sources.

Current status (as of August 2026): The preprint claims the cavity asymptotics problem is solved, but its proof remains unverified; absent this claim, the problem remains open.

Sources

arxiv.org

No solutions have been posted yet.