# P3096: Persistent exponential stretching of a material line in two-dimensional Euler flow

- ID: `P3096`
- Reference: `euler-material-line-exponential-growth`
- Page: https://theoremdb.org/statements/P3096
- Record maturity: Reviewed problem with recorded work

## Problem

Does there exist a smooth bounded planar domain, a smooth global solution of the two-dimensional incompressible Euler equation in that domain, and an initial unit line segment transported by the Lagrangian flow whose length is at least \(c e^{ct}\) for every \(t\ge0\) and some \(c>0\)?

### Context

Known frontier: Very fast vorticity-gradient growth and substantial finite-time material deformation are known.

Open boundary: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record.

### Problem setup

- **Definition (Lagrangian flow).** Φ_t solves dΦ_t(x)/dt=u(Φ_t(x),t) with Φ_0(x)=x.
- **Definition (material line).** The curve Φ_t(I) obtained from an initial segment I.
- **Remark.** The segment is made of fluid particles. Its later image can bend, and its arclength measures persistent stretching rather than a transient gradient spike.

### What counts as a solution

- Construct a domain and smooth Euler solution with the stated all-time arclength lower bound.
- Or prove an obstruction excluding such all-time exponential stretching.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: Very fast vorticity-gradient growth and substantial finite-time material deformation are known. Exact unresolved remainder: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record. [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: Very fast vorticity-gradient growth and substantial finite-time material deformation are known. Exact unresolved remainder: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record.

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

The strongest neighboring result found in the cited sources is: Very fast vorticity-gradient growth and substantial finite-time material deformation are known.

The exact unresolved remainder is: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record.

A complete resolution must meet the following acceptance conditions:
- Construct a domain and smooth Euler solution with the stated all-time arclength lower bound.
- Or prove an obstruction excluding such all-time exponential stretching.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: Very fast vorticity-gradient growth and substantial finite-time material deformation are known. Exact unresolved remainder: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: 2D Euler material line length exponential for all time; Euler transported line segment exponential stretching bounded domain
- Strongest checked neighboring result: Very fast vorticity-gradient growth and substantial finite-time material deformation are known.
- Exact unresolved remainder: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record.

### Other known results

- **Claim 2** (supported): Very fast vorticity-gradient growth and substantial finite-time material deformation are known. [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: Very fast vorticity-gradient growth and substantial finite-time material deformation are known. Unresolved remainder: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `euler-material-line-exponential-growth`, 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 4 https://aimath.org/pastworkshops/smallscalefluidrep.pdf
   - Also cited at AIM, Small scales and singularity formation in fluid dynamics, workshop report, Problem 4. Problem 4
   - website; reference source; checked 2026-08-01
   - Source use: original_summary
   - States the unit-segment exponential-growth problem.
   - Source used to assess the problem's recorded status.
   - For Persistent exponential stretching of a material line in two-dimensional Euler flow: This is the dated publication status for the canonical target Persistent exponential stretching of a material line in two-dimensional Euler flow.
   - Source named by the research packet.
2. <a id="reference-2"></a>Alexander Kiselev and Vladimir Šverák, “Small scale creation for solutions of the incompressible two-dimensional Euler equation”. Annals of Mathematics (2014), 1205-1220. DOI 10.4007/annals.2014.180.3.9. Theorem 1.1 https://doi.org/10.4007/annals.2014.180.3.9
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Constructs double-exponential gradient growth in a bounded domain, demonstrating strong boundary-driven stretching mechanisms.
   - Source used to assess the problem's recorded status.
   - For Persistent exponential stretching of a material line in two-dimensional Euler flow: Constructs double-exponential gradient growth in a bounded domain, demonstrating strong boundary-driven stretching mechanisms.