Pith Number

pith:C6JH6NCB

pith:2026:C6JH6NCBZNYX7ASH26GT2WO3TS

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Mixing and Small-Scale Formation in a Passive Divergence-Free Vector Field

Anuj Kumar, Franziska Weber

Numerical simulations indicate that passive divergence-free vector fields mix at least exponentially when the advecting field is chosen at each instant to maximize instantaneous decay of the negative Sobolev norm.

arxiv:2605.12883 v1 · 2026-05-13 · math.AP

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

Numerical simulations provide evidence that the optimal mixing rate is at least exponential in time.

C2weakest assumption

That a divergence-free U with bounded W^{1,q} norm exists at each instant which instantaneously maximizes the decay of the H^{-α} norm of u, and that the resulting evolution remains well-defined.

C3one line summary

Derives lower bounds on mixing rates for passive divergence-free vector fields under W^{1,q} constraints and provides numerical evidence for at least exponential optimal mixing via H^{-α} norm decay.

References

34 extracted · 34 resolved · 0 Pith anchors

[1] G. Alberti, G. Crippa, and A. L. Mazzucato. Exponential self-similar mixing by incompressible flows.J. Amer. Math. Soc., 32(2):445–490, 2019 2019

[2] A. Bressan. A lemma and a conjecture on the cost of rearrangements.Rendiconti del Seminario Matematico della Universita di Padova, 110:97–102, 2003 2003

[3] H. Brezis and P. Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35(5):1355–1376, 2018. 24 Mixing and Small-Scale For 2018

[4] H. Brezis and P. Mironescu. Where Sobolev interacts with Gagliardo-Nirenberg.J. Funct. Anal., 277(8):2839–2864, 2019 2019

[5] E. Bru` e, M. Colombo, and A. Kumar. Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field.arXiv preprint arXiv:2405.01670, 2024 2024

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T03:09:11.033578Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

17927f3441cb717f8247d78d3d59db9cbb3443339426cf3f99ddef603145fff5

Aliases

arxiv: 2605.12883 · arxiv_version: 2605.12883v1 · doi: 10.48550/arxiv.2605.12883 · pith_short_12: C6JH6NCBZNYX · pith_short_16: C6JH6NCBZNYX7ASH · pith_short_8: C6JH6NCB

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/C6JH6NCBZNYX7ASH26GT2WO3TS \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 17927f3441cb717f8247d78d3d59db9cbb3443339426cf3f99ddef603145fff5

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "002c0e91c278888340a876b02fce17c06c8521b744ccf821f17216111d17af59",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-05-13T01:57:33Z",
    "title_canon_sha256": "9fa5f56df8d8a9019e80c6ccc3d6c1e92388a991f4d5535ffe2372e37465c774"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12883",
    "kind": "arxiv",
    "version": 1
  }
}