[#P3122] Backward self-similar Navier-Stokes profiles in a half-space
Contents
Problem. Let \(U\) and \(P\) solve \(-\Delta U-\tfrac{1}{2}U-\tfrac{1}{2}(x\cdot\nabla)U+(U\cdot\nabla)U+\nabla P=0\) and \(\nabla\cdot U=0\) in the three-dimensional upper half-space, with \(U=0\) on the boundary. Under scale-critical integrability such as \(U\in L^3\), must \(U\) vanish identically?
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Definitions and notation
1Context
Known frontier: Tsai-type theorems rule out broad classes of whole-space profiles, and half-space results exist under stronger decay or regularity. Open boundary: The scale-critical half-space statement above remains open in the checked sources.
2Problem setup
Definition 1 (Leray profile). The time-independent profile U in u(x,t)=(-t)^(-1/2)U(x/√(-t)).
Definition 2 (scale-critical). A norm unchanged by the Navier-Stokes scaling, including L³ in three dimensions.
Remark 1. A nonzero profile would generate a backward self-similar Navier-Stokes solution and hence a candidate singularity at one time. The half-space boundary makes pressure and reflection arguments harder than in the whole space.
3What counts as a solution
- Prove U=0 under a precisely stated critical hypothesis no stronger than the packet's target.
- Or construct a nonzero profile meeting all equations, boundary conditions, and integrability assumptions.
1Status
Saved packet · August 1, 2026
Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Tsai-type theorems rule out broad classes of whole-space profiles, and half-space results exist under stronger decay or regularity. Exact unresolved remainder: The scale-critical half-space statement above remains open in the checked sources.[1][2]
1Packet records
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Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Tsai-type theorems rule out broad classes of whole-space profiles, and half-space results exist under stronger decay or regularity. Exact unresolved remainder: The scale-critical half-space statement above remains open in the checked sources.
- Equivalent-formulation queries: half-space backward self-similar Navier Stokes L3 nonexistence; Leray profile upper half space no slip open problem
- Strongest checked neighboring result: Tsai-type theorems rule out broad classes of whole-space profiles, and half-space results exist under stronger decay or regularity.
- Exact unresolved remainder: The scale-critical half-space statement above remains open in the checked sources.
How the 4 records connect
ProblemBackward self-similar Navier-Stokes profiles in a half-space
All 3 recorded relations between these records and the problem
2See also
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“Backward self-similar Navier-Stokes profiles in a half-space.” TheoremDB. P3122. Problem statement; statement text SHA-256 1f85bdbb630299d163b047da8c6fee41c15489776ca07e174a788dd0cbcf0eeb. https://theoremdb.org/statement/?ref=P3122
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note = {Problem statement; statement text SHA-256 1f85bdbb630299d163b047da8c6fee41c15489776ca07e174a788dd0cbcf0eeb},
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This problem includes 4 records joined by 3 typed links, sourced from aimath.org[1], current as of August 1, 2026.
1References
- Packet source. Tarek Elgindi, Aseel Farhat, Anna Mazzucato, and Wojciech Ożański, organizers, “Small Scale Dynamics in Incompressible Fluid Flows,” AIM workshop summary, checked 2026-08-01. Problem 2. ↗website · reference source · checked 2026-08-01Source use: original summary.Asks for a half-space extension of Tsai-type nonexistence under weaker decay.Also cited at AIM, Small scales and singularity formation in fluid dynamics, workshop report, Problem 2. Problem 2.Source used to assess the problem's recorded status.For Backward self-similar Navier-Stokes profiles in a half-space: This is the dated publication status for the canonical target Backward self-similar Navier-Stokes profiles in a half-space.Source named by the research packet.
- Tai-Peng Tsai, “On Leray's Self-Similar Solutions of the Navier-Stokes Equations Satisfying Local Energy Estimates”. Archive for Rational Mechanics and Analysis 143(1) (1998), 29-51. DOI 10.1007/s002050050099. main nonexistence theorems. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supplies the classical whole-space nonexistence framework that the half-space problem seeks to extend.Source used to assess the problem's recorded status.For Backward self-similar Navier-Stokes profiles in a half-space: Supplies the classical whole-space nonexistence framework that the half-space problem seeks to extend.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.
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