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

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 \(c>0\)?

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1Context

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.

2Problem setup

Definition 1 (Lagrangian flow). Φ_t solves dΦ_t(x)/dt=u(Φ_t(x),t) with Φ_0(x)=x.

Definition 2 (material line). The curve Φ_t(I) obtained from an initial segment I.

Remark 1. The segment is made of fluid particles. Its later image can bend, and its arclength measures persistent stretching rather than a transient gradient spike.

3What 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.

1Status

What counts as a solution

Current status (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.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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][2] Relevance to this problem This is the dated publication status for the canonical target Persistent exponential stretching of a material line in two-dimensional Euler flow. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details 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. Open full record →
Very fast vorticity-gradient growth and substantial finite-time material deformation are known.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Very fast vorticity-gradient growth and substantial finite-time material deformation are known.^[1][2] Relevance to this problem Locates the present research frontier immediately below Persistent exponential stretching of a material line in two-dimensional Euler flow. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Very fast vorticity-gradient growth and substantial finite-time material deformation are known. This leaves the following boundary unresolved: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
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.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 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][2] Relevance to this problem Documents why Persistent exponential stretching of a material line in two-dimensional Euler flow was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - 2D Euler material line length exponential for all time - Euler transported line segment exponential stretching bounded domain 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. Related: Claim 1 Open full record →
A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record. Relevance to this problem Turns the remaining uncertainty in Persistent exponential stretching of a material line in two-dimensional Euler flow into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: A single smooth example with a uniform exponential arclength lower bound for every time remains open in the checked record. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - 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. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion material

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.

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.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemPersistent exponential stretching of a material line in two-dimensional Euler flow

All 3 recorded relations between these records and the problem

2See also

Navier-Stokes existence and smoothnessanalysis
Three-dimensional Euler regularityanalysis
Bochner-Riesz conjecture in higher dimensionsanalysis

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Plain text

“Persistent exponential stretching of a material line in two-dimensional Euler flow.” TheoremDB. P3096. Problem statement; statement text SHA-256 f27e89f4d0cdc81a1733ebeea8d8bcb45da25586f66c6f3946c013e1871dee99. https://theoremdb.org/statement/?ref=P3096

BibTeX

@misc{theoremdb-problem-f27e89f4d0cdc81a1733ebeea8d8bcb45da25586f66c6f3946c013e1871dee99,
  title = {{Persistent exponential stretching of a material line in two-dimensional Euler flow}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f27e89f4d0cdc81a1733ebeea8d8bcb45da25586f66c6f3946c013e1871dee99},
  url = {https://theoremdb.org/statement/?ref=P3096}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from aimath.org^[1], current as of August 1, 2026.

1Lean verification

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 4. ↗website · reference source · checked 2026-08-01Source use: original summary.States the unit-segment exponential-growth problem.Also cited at AIM, Small scales and singularity formation in fluid dynamics, workshop report, Problem 4. Problem 4.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.
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. ↗journal article · primary source · checked 2026-08-01Source 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.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

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

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