# P3122: Backward self-similar Navier-Stokes profiles in a half-space

- ID: `P3122`
- Reference: `navier-stokes-half-space-backward-self-similar`
- Page: https://theoremdb.org/statements/P3122
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (Leray profile).** The time-independent profile U in u(x,t)=(-t)^(-1/2)U(x/√(-t)).
- **Definition (scale-critical).** A norm unchanged by the Navier-Stokes scaling, including L³ in three dimensions.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: Tsai-type theorems rule out broad classes of whole-space profiles, and half-space results exist under stronger decay or regularity.

The exact unresolved remainder is: The scale-critical half-space statement above remains open in the checked sources.

A complete resolution must meet the following acceptance conditions:
- 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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): Tsai-type theorems rule out broad classes of whole-space profiles, and half-space results exist under stronger decay or regularity. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Tsai-type theorems rule out broad classes of whole-space profiles, and half-space results exist under stronger decay or regularity. Unresolved remainder: The scale-critical half-space statement above remains open in the checked sources. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): The scale-critical half-space statement above remains open in the checked sources.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `navier-stokes-half-space-backward-self-similar`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://aimath.org/pastworkshops/smallscalefluidrep.pdf
   - Also cited at AIM, Small scales and singularity formation in fluid dynamics, workshop report, Problem 2. Problem 2
   - website; reference source; checked 2026-08-01
   - Source use: original_summary
   - Asks for a half-space extension of Tsai-type nonexistence under weaker decay.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1007/s002050050099
   - journal_article; primary source; checked 2026-08-01
   - Source 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.