Oleinik–Samokhin Open Problem 5 on stationary Prandtl separation

For the two-dimensional stationary Prandtl system under a constant adverse pressure gradient, determine the local asymptotic structure of a solution near a finite separation point $x=x^*$ , where the wall shear $$\partial_y u(x,0) $tends to zero as$ x\uparrow x^ $. In particular, classify the possible separation laws and determine which exponents and asymptotic profiles can occur in expressions of the form$ \partial_y u(x,0)\sim C(x^-x)^\alpha $, with$ C>0$, together with the corresponding local expansions of the solution.

References

Primary source

arXiv

Additional references

Local separation profiles for the stationary Prandtl system — arXiv — Ky Giao Duong, Tej-Eddine Ghoul, Sameer Iyer, Yasunori Maekawa

fluid-dynamics partial-differential-equations

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

A new paper reports that every integer-strength local separation profile can occur, substantially extending earlier partial results without settling the wider separation problem.

Oleinik and Samokhin asked for the local structure of stationary Prandtl flow near a boundary-layer separation point. Earlier work established separation and specific asymptotic rates, but not a complete classification.

Known results

Dalibard and Masmoudi (2018): for special data and $p'(x)=1$ , $\partial_y u(x,0)\sim C\sqrt{x^*-x}$ .
Duong, Ghoul, Iyer, and Maekawa (2019): finite-time separation for a large class of data and partial local information, including $\partial_y u(x,0)\le C(x^*-x)^{1/4}$ .

October 2026 classification claim

Duong, Ghoul, Iyer, and Maekawa report smooth data realizing exponent $\ell/2$ for every integer $\ell\ge1$ , together with local asymptotics and stability analysis for $\ell=1$ . This is substantial claimed progress on Open Problem 5, but the retrieved record does not independently verify the result.

Current status (as of October 2026): Earlier partial results are established, while the all- $\ell/2$ classification is a claimed but unverified advance; the broader Prandtl separation questions remain open.

Sources

No solutions have been posted yet.