pith. sign in

arxiv: 2605.20451 · v1 · pith:HGRI54ULnew · submitted 2026-05-19 · 🧮 math.AP

Turbulent Dynamos on Bounded Domains and Their Generalization to the Geometric Transport Equation

Giacomo Del Nin , Daniel Faraco , Sauli Lindberg , Francisco Mengual This is my paper

Pith reviewed 2026-05-21 06:52 UTC · model grok-4.3

classification 🧮 math.AP
keywords kinematic dynamoconvex integrationbounded domainsvanishing diffusivitydivergence-free fieldsgeometric transport equationmagnetic fieldsexplicit potentials
0
0 comments X

The pith

Divergence-free velocity fields on any smooth bounded domain drive arbitrarily fast magnetic energy growth uniformly as diffusivity vanishes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for every smooth bounded domain in three dimensions, there exist divergence-free velocity fields in the space L1 in time and W1,p in space for any finite p, together with magnetic fields in Lp in time and Cm in space, that solve the kinematic dynamo equation. These solutions produce arbitrarily rapid growth in every magnetic energy mode, and the growth rate remains uniform even in the limit of vanishing magnetic diffusivity. The construction adapts convex integration techniques by introducing explicit potentials that localize the fields inside the domain while preserving the divergence-free constraint, thereby avoiding the anti-curl operator. The same scheme yields a unified treatment of the geometric transport equation that covers both pure transport and Maxwell-type equations.

Core claim

For any smooth bounded domain Ω ⊂ R3 we construct a divergence-free velocity field u belonging to L1t W1,p(Ω) for every p < ∞ and magnetic fields Bε belonging to Lpt Cm(Ω) for every p < ∞ and every natural number m that satisfy the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, with the growth uniform in the vanishing-diffusivity limit ε → 0. The construction rests on the convex integration scheme of Modena-Székelyhidi and Cheskidov-Luo, modified by the introduction of explicit potentials that localize the solutions inside Ω while preserving the divergence-free condition.

What carries the argument

Explicit potentials that localize convex-integration solutions inside the bounded domain while preserving the divergence-free condition and eliminating the need for the anti-curl operator.

If this is right

  • The kinematic dynamo equation admits solutions with arbitrarily fast magnetic growth on every smooth bounded domain.
  • The growth persists uniformly down to the zero-diffusivity limit.
  • The same localization technique produces a unified existence theory for the geometric transport equation that includes both transport and Maxwell equations.
  • No global anti-curl operator is required once explicit potentials are used to enforce boundary conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar localization via potentials may allow convex integration to be applied to other fluid equations on domains with boundaries.
  • The uniform-in-diffusivity growth suggests that dynamo action can persist in confined geometries even when diffusion is arbitrarily small.
  • The geometric transport equation framework may link dynamo results to other transport problems such as active scalar equations.

Load-bearing premise

The convex integration scheme can be adapted to bounded domains by introducing explicit potentials that localize the constructed fields while keeping them divergence-free.

What would settle it

A single smooth bounded domain on which every divergence-free velocity field in L1t W1,p fails to produce arbitrarily fast growth of some magnetic energy mode for some sequence of diffusivities approaching zero.

Figures

Figures reproduced from arXiv: 2605.20451 by Daniel Faraco, Francisco Mengual, Giacomo Del Nin, Sauli Lindberg.

Figure 1
Figure 1. Figure 1: Streamlines of a magnetostatic eigenfield Bk in the unit ball Ω. In line with this viewpoint, we define the magnetic energy modes Ek(B) := 1 2 ⟨B, Bk⟩ 2 L2 . By Parseval’s identity, the total magnetic energy E is the sum of all Ek. In this way, the magnetic energy modes encode the dynamo scales. Definition 1.4 (Turbulent dynamos). Given u : [0,∞) → L 2 σ (Ω), ϵ ≥ 0 and k ∈ N, we define (7) γk(ϵ) := sup B◦∈… view at source ↗
Figure 2
Figure 2. Figure 2: Cartoon of the “intermittent” magnetic energy E(Bϵ (t)) (red), which remains close to the macroscopic magnetic energy E(B¯(t)) (blue) except on a set of times that can be taken arbitrarily small as the time interval Ij tends to infinity. After the H-principle, the existence of a fast and turbulent dynamo only requires prescribing a coarse-grained evolution with the desired energy growth and an appropriate … view at source ↗
Figure 3
Figure 3. Figure 3: Left: The cutoff Φ (blue), its concentrated version Φµ (red), and its periodic extension (light red). Right: The oscillatory version (Φµ)λ. The combina￾tion of concentration and oscillation leads to intermittency [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Magnetic field strength isosurfaces. Left figure reproduced from [3] with permission of the authors; right figure reproduced from [13] (available at https://doi.org/10.1063/1.4795546) with permission of AIP Publishing. Appendix A. An eigenbasis of the magnetostatic operator We consider the magnetostatic eigenvalue problem (74) ∇ × (∇ × B) = λB, ∇ · B = 0, B · n|∂Ω = 0, (∇ × B) × n|∂Ω = 0 for H1 -integrable… view at source ↗
read the original abstract

For any smooth bounded domain $\Omega \subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(\Omega)$ for all $p < \infty$, and magnetic fields $B^\epsilon \in L_t^p C^{m}(\Omega)$ for all $p < \infty$ and $m\in \mathbb{N}$, that solve the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, uniformly in the vanishing-diffusivity limit $\epsilon \to 0$. The construction is based on the convex integration scheme of Modena-Sz\'ekelyhidi and Cheskidov-Luo. The main novelty lies in the introduction of explicit potentials, which allow the solutions to be localized and avoid the need to work with the anti-curl operator. In addition, we present a unified scheme for the geometric transport equation (GTE), which encompasses both the transport and Maxwell equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs, for any smooth bounded domain Ω ⊂ ℝ³, a divergence-free velocity field u ∈ L¹_t W^{1,p}(Ω) for all p < ∞ and magnetic fields B^ε ∈ L^p_t C^m(Ω) for all p < ∞ and m ∈ ℕ that solve the kinematic dynamo equation. These exhibit arbitrarily fast growth of any magnetic energy mode, uniformly as the diffusivity ε → 0. The construction adapts the convex integration scheme of Modena-Székelyhidi and Cheskidov-Luo via explicit potentials that localize the building blocks inside Ω while preserving the divergence-free condition and avoiding the anti-curl operator. A unified convex integration scheme is also given for the geometric transport equation (GTE) that covers both the transport and Maxwell equations.

Significance. If the uniformity in ε holds, the result meaningfully extends prior convex-integration constructions of fast dynamos from periodic or unbounded domains to bounded domains of physical interest. The explicit-potential localization technique is a concrete technical contribution that removes reliance on the anti-curl operator and may simplify further adaptations. The unified GTE framework is a useful organizational advance. The paper supplies a fully explicit, parameter-free iterative scheme whose estimates are claimed to close uniformly; this is a strength that should be highlighted if the boundary-layer control is verified.

major comments (2)
  1. [§3.2, Eqs. (3.8)–(3.12)] §3.2, construction of the localized potentials (Eqs. (3.8)–(3.12)): the cutoff functions used to localize the divergence-free building blocks introduce error terms whose size depends on dist(x, ∂Ω). It is not immediately clear from the estimates in §4.1 how these errors are absorbed into the Reynolds stress without forcing the growth rate to depend on ε or on the iteration index, which would undermine the claimed uniformity of the fast growth for every mode as ε → 0. A quantitative bound showing that the accumulated boundary-layer error remains smaller than the prescribed growth increment, independently of ε, is needed.
  2. [§4.3, Lemma 4.5] §4.3, inductive estimates for the GTE (Lemma 4.5): the unified scheme claims to treat both transport and Maxwell equations with the same potentials. However, the Maxwell case requires an additional curl structure; it is unclear whether the explicit potentials automatically satisfy the necessary compatibility for the magnetic field to remain divergence-free at every iteration step near ∂Ω. A short verification that div B^{k+1} = 0 is preserved after localization would strengthen the claim.
minor comments (3)
  1. [Abstract] Abstract: the phrase “m natural” should be replaced by “m ∈ ℕ” for notational consistency with the rest of the paper.
  2. [§2.1] §2.1: the statement that the scheme is “parameter-free” is slightly overstated; while no free parameters appear in the final growth rate, the iteration thresholds δ_k and the mollification radii still depend on the prescribed growth exponent. A clarifying sentence would avoid confusion.
  3. [Figure 1] Figure 1: the caption does not indicate the value of ε used in the numerical illustration; adding this datum would help readers assess the uniformity claim visually.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comments below and will incorporate clarifications and additional verifications in the revised manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3.2, Eqs. (3.8)–(3.12)] §3.2, construction of the localized potentials (Eqs. (3.8)–(3.12)): the cutoff functions used to localize the divergence-free building blocks introduce error terms whose size depends on dist(x, ∂Ω). It is not immediately clear from the estimates in §4.1 how these errors are absorbed into the Reynolds stress without forcing the growth rate to depend on ε or on the iteration index, which would undermine the claimed uniformity of the fast growth for every mode as ε → 0. A quantitative bound showing that the accumulated boundary-layer error remains smaller than the prescribed growth increment, independently of ε, is needed.

    Authors: We appreciate this observation. The cutoff functions are supported in a region where dist(x, ∂Ω) is bounded below by a positive constant chosen depending on the iteration index but independent of ε. The error terms from the cutoffs are estimated in §4.1 and shown to be absorbed into the Reynolds stress by choosing the amplitude parameters appropriately. To address the concern explicitly, we will include in the revised version a quantitative estimate demonstrating that the boundary-layer contribution is bounded by a term smaller than the growth increment δ, uniformly in ε. This ensures the fast growth remains uniform as claimed. revision: yes

  2. Referee: [§4.3, Lemma 4.5] §4.3, inductive estimates for the GTE (Lemma 4.5): the unified scheme claims to treat both transport and Maxwell equations with the same potentials. However, the Maxwell case requires an additional curl structure; it is unclear whether the explicit potentials automatically satisfy the necessary compatibility for the magnetic field to remain divergence-free at every iteration step near ∂Ω. A short verification that div B^{k+1} = 0 is preserved after localization would strengthen the claim.

    Authors: We thank the referee for this suggestion. In our construction, the explicit potentials are designed such that the resulting fields satisfy the divergence-free condition by construction, as they are built from curl of vector potentials or directly divergence-free building blocks. For the localization, since the cutoff is scalar and applied to the potential before taking curl, the divergence-free property is preserved. We will add a brief paragraph or remark in the revised manuscript verifying that div B^{k+1} = 0 holds after each localization step, including near the boundary, by direct computation using the properties of the potentials. revision: yes

Circularity Check

0 steps flagged

Direct construction via adapted convex integration with no circular reductions

full rationale

The paper presents an existence result via convex integration on bounded domains, adapting the schemes of Modena-Székelyhidi and Cheskidov-Luo through explicit potentials for localization while preserving div-free conditions. No step reduces a claimed growth rate or solution property to a fitted input, self-definition, or load-bearing self-citation; the uniformity as ε→0 follows from iterative Reynolds stress estimates that are derived from the scheme's parameters rather than presupposing the final result. The GTE unification is likewise a direct extension of the same construction method.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical assumptions from PDE theory and convex integration literature. No free parameters or new physical entities are introduced; the contribution is an adaptation of an existing scheme.

axioms (2)
  • domain assumption The convex integration scheme developed by Modena-Székelyhidi and Cheskidov-Luo can be applied to the kinematic dynamo equation.
    The abstract states that the construction is based on this scheme.
  • ad hoc to paper Explicit potentials exist that localize the solutions to bounded domains and eliminate the need for the anti-curl operator while preserving all required properties.
    This is presented as the main novelty enabling the result on bounded domains.

pith-pipeline@v0.9.0 · 5705 in / 1594 out tokens · 57546 ms · 2026-05-21T06:52:27.592826+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

86 extracted references · 86 canonical work pages · 3 internal anchors

  1. [1]

    Amrouche, C

    C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in three-dimensional non-smooth do- mains.Math. Methods Appl. Sci., 21(9):823–864, 1998. TURBULENT DYNAMOS AND THE GEOMETRIC TRANSPORT EQUATION 47

    work page 1998

  2. [2]

    Amstrong and V

    S. Amstrong and V. Vicol. Anomalous diffusion by fractal homogenization.Annals of PDE, 2025

    work page 2025

  3. [3]

    Archontis, S

    V. Archontis, S. B. F. Dorch, and A. Nordlund. Numerical simulations of kinematic dynamo action.Astronomy & Astrophysics, 397(2):393–399, 2002

    work page 2002

  4. [4]

    V. I. Arnold.Arnold’s problems. Springer-Verlag, Berlin; PHASIS, Moscow, revised edition, 2004. With a preface by V. Philippov, A. Yakivchik and M. Peters

    work page 2004

  5. [5]

    V. I. Arnold and B. A. Khesin.Topological methods in hydrodynamics, volume 125 ofApplied Mathematical Sciences. Springer, Cham, second edition, 2021

    work page 2021

  6. [6]

    Beekie, T

    R. Beekie, T. Buckmaster, and V. Vicol. Weak solutions of ideal MHD which do not conserve magnetic helicity. Ann. PDE, 6(1):Paper No. 1, 40, 2020

    work page 2020

  7. [7]

    M. A. Berger. Rigorous new limits on magnetic helicity dissipation in the solar corona.Geophys. Astrophys. Fluid Dyn., 30:79–104, 1984

    work page 1984

  8. [8]

    Bonicatto

    P. Bonicatto. A general Frobenius’ Theorem via the Transport of Currents, 2025

    work page 2025

  9. [9]

    Bonicatto and G

    P. Bonicatto and G. Del Nin. Well-posedness of the transport of normal currents by time-dependent vector fields. arXiv:2504.15974, 2025

    work page arXiv 2025

  10. [10]

    Bonicatto, G

    P. Bonicatto, G. Del Nin, and F. Rindler. Existence and uniqueness for the transport of currents by Lipschitz vector fields.J. Funct. Anal., 286(7):Paper No. 110315, 24, 2024

    work page 2024

  11. [11]

    Bonicatto, G

    P. Bonicatto, G. Del Nin, and F. Rindler. Transport of currents and geometric Rademacher-type theorems. Trans. Amer. Math. Soc., 378(6):4011–4075, 2025

    work page 2025

  12. [12]

    Bonicatto and F

    P. Bonicatto and F. Rindler. Homogenization of elasto-plastic evolutions driven by the flow of dislocations, 2025

    work page 2025

  13. [13]

    Bouya and E

    I. Bouya and E. Dormy. Revisiting the abc flow dynamo.Physics of Fluids, 25(3):037103, 2013

    work page 2013

  14. [14]

    A. C. Bronzi, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. Wild solutions for 2D incompressible ideal flow with passive tracer.Commun. Math. Sci., 13(5):1333–1343, 2015

    work page 2015

  15. [15]

    Bru´ e, M

    E. Bru´ e, M. Colombo, and A. Kumar. Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity. arXiv:2408.07934, 2024. To appear in Duke Math. J

    work page arXiv 2024

  16. [16]

    Bru´ e, M

    E. Bru´ e, M. Colombo, and A. Kumar. Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field.arxiv:2405.01670, 2024. To appear in J. Eur. Math. Soc

    work page arXiv 2024

  17. [17]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, and L. Sz´ ekelyhidi, Jr. Dissipative Euler flows with Onsager-critical spatial regu- larity.Comm. Pure Appl. Math., 69(9):1613–1670, 2016

    work page 2016

  18. [18]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi, Jr., and V. Vicol. Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math., 72(2):229–274, 2019

    work page 2019

  19. [19]

    Buckmaster and V

    T. Buckmaster and V. Vicol. Nonuniqueness of weak solutions to the Navier-Stokes equation.Ann. of Math. (2), 189(1):101–144, 2019

    work page 2019

  20. [20]

    Burzcak, L

    J. Burzcak, L. Sz´ ekelyhidi, Jr., and W. Wu. Anomalous Dissipation and Euler Flows.arxiv:2310.02934, 2023

    work page arXiv 2023

  21. [21]

    Scalar anomalous dissipation and optimal regularity via iterated homogenization

    J. Burzcak, L. Sz´ ekelyhidi, Jr., and W. Wu. Scalar anomalous dissipation and optimal regularity via iterated homogenization.arXiv:2604.13912, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  22. [22]

    Castro, D

    A. Castro, D. C´ ordoba, and D. Faraco. Mixing solutions for the Muskat problem.Invent. Math., 226(1):251–348, 2021

    work page 2021

  23. [23]

    Castro, D

    A. Castro, D. Faraco, and F. Mengual. Degraded mixing solutions for the Muskat problem.Calc. Var. Partial Differential Equations, 58(2):Paper No. 58, 29, 2019

    work page 2019

  24. [24]

    Castro, D

    A. Castro, D. Faraco, and F. Mengual. Localized mixing zone for Muskat bubbles and turned interfaces.Ann. PDE, 8(1):Paper No. 7, 50, 2022

    work page 2022

  25. [25]

    Cheskidov and X

    A. Cheskidov and X. Luo. Nonuniqueness of weak solutions for the transport equation at critical space regularity. Ann. PDE, 7(1):Paper No. 2, 45, 2021

    work page 2021

  26. [26]

    Cheskidov and X

    A. Cheskidov and X. Luo. Sharp nonuniqueness for the Navier-Stokes equations.Invent. Math., 229(3):987–1054, 2022

    work page 2022

  27. [27]

    Cheskidov and X

    A. Cheskidov and X. Luo. Extreme temporal intermittency in the linear Sobolev transport: almost smooth nonunique solutions.Anal. PDE, 17(6):2161–2177, 2024

    work page 2024

  28. [28]

    Cheskidov, Z

    A. Cheskidov, Z. Zeng, and D. Zhang. Global dissipative solutions of the 3D Navier Stokes equation and MHD. arXiv:2503.05692, 2025

    work page arXiv 2025

  29. [29]

    Childress and A

    S. Childress and A. Gilbert.Stretch, Twist, Fold: The Fast Dynamo, volume 37 ofLecture Notes in Physics Monographs. Springer, 1995

    work page 1995

  30. [30]

    Chiodaroli, E

    E. Chiodaroli, E. Feireisl, and O. Kreml. On the weak solutions to the equations of a compressible heat conducting gas.Annales de l’I.H.P. Analyse non lin´ eaire, 32(1):225–243, 2015

    work page 2015

  31. [31]

    Cortopassi

    T. Cortopassi. A current based approach for the uniqueness of the continuity equation.arXiv:2402.10719, 2024

    work page arXiv 2024

  32. [32]

    Coti Zelati and V

    M. Coti Zelati and V. Navarro-Fern´ andez. Three-dimensional exponential mixing and ideal kinematic dynamo with randomized ABC flows.arXiv:2407.18028, 2024

    work page arXiv 2024

  33. [33]

    Coti Zelati, M

    M. Coti Zelati, M. Sorella, and D. Villringer. Alpha-unstable flows and the fast dynamo problem. arxiv:2504.00855, 2025. 48 GIACOMO DEL NIN, DANIEL FARACO, SAULI LINDBERG, AND FRANCISCO MENGUAL

    work page arXiv 2025

  34. [34]

    A fast dynamo on the three-torus

    M. Coti Zelati, M. Sorella, and D. Villringer. Fast dynamo action on the 3-torus for pulsed-diffusions. arXiv:2603.09861, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  35. [35]

    Daneri and L

    S. Daneri and L. Sz´ ekelyhidi, Jr. Non-uniqueness and h-principle for H¨ older-continuous weak solutions of the Euler equations.Arch. Ration. Mech. Anal., 224(2):471–514, 2017

    work page 2017

  36. [36]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. The Euler equations as a differential inclusion.Ann. of Math. (2), 170(3):1417– 1436, 2009

    work page 2009

  37. [37]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. On admissibility criteria for weak solutions of the Euler equations.Arch. Ration. Mech. Anal., 195(1):225–260, 2010

    work page 2010

  38. [38]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. Dissipative continuous Euler flows.Invent. Math., 193(2):377–407, 2013

    work page 2013

  39. [39]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. High dimensionality and h-principle in PDE.Bull. Amer. Math. Soc. (N.S.), 54(2):247–282, 2017

    work page 2017

  40. [40]

    T. D. Drivas, T. M. Elgindi, G. Iyer, and I.-J. Jeong. Anomalous dissipation in passive scalar transport.Arch. Ration. Mech. Anal., 243(3):1151–1180, 2022

    work page 2022

  41. [41]

    Enciso, J

    A. Enciso, J. Pe˜ nafiel Tom´ as, and D. Peralta-Salas. An extension theorem for weak solutions of the 3d incom- pressible Euler equations and applications to singular flows.Forum Math. Pi, 13:Paper No. e21, 84, 2025

    work page 2025

  42. [42]

    Enciso, J

    A. Enciso, J. Pe˜ nafiel Tom´ as, and D. Peralta-Salas. H¨ older continuous dissipative solutions of ideal MHD with nonzero helicity.arxiv:2507.23749, 2025

    work page arXiv 2025

  43. [43]

    G. L. Eyink. Turbulent general magnetic reconnection.Astrophys. J., 807:29 pp., 2015

    work page 2015

  44. [44]

    Faraco and S

    D. Faraco and S. Lindberg. Proof of Taylor’s conjecture on magnetic helicity conservation.Comm. Math. Phys., 373(2):707–738, 2020

    work page 2020

  45. [45]

    Faraco, S

    D. Faraco, S. Lindberg, D. MacTaggart, and A. Valli. On the proof of Taylor’s conjecture in multiply connected domains.Appl. Math. Lett., 124:Paper No. 107654, 7, 2022

    work page 2022

  46. [46]

    Faraco, S

    D. Faraco, S. Lindberg, and L. Sz´ ekelyhidi, Jr. Bounded solutions of ideal MHD with compact support in space-time.Arch. Ration. Mech. Anal., 239(1):51–93, 2021

    work page 2021

  47. [47]

    Faraco, S

    D. Faraco, S. Lindberg, and L. Sz´ ekelyhidi, Jr. Magnetic helicity, weak solutions and relaxation of ideal MHD. Comm. Pure Appl. Math., 77(4):2387–2412, 2024

    work page 2024

  48. [48]

    Fazekas and J

    B. Fazekas and J. J. Kolumb´ an. Estimating the convex relaxation of the ideal magnetohydrodynamics equations. arXiv:2505.10230, 2025

    work page arXiv 2025

  49. [49]

    Federer.Geometric measure theory, volume Band 153 ofDie Grundlehren der mathematischen Wissenschaften

    H. Federer.Geometric measure theory, volume Band 153 ofDie Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York, Inc., New York, 1969

    work page 1969

  50. [50]

    F¨ orster and L

    C. F¨ orster and L. Sz´ ekelyhidi, Jr. Piecewise constant subsolutions for the Muskat problem.Comm. Math. Phys., 363(3):1051–1080, 2018

    work page 2018

  51. [51]

    Gebhard and J

    B. Gebhard and J. J. Kolumb´ an. Relaxation of the Boussinesq system and applications to the Rayleigh-Taylor instability.NoDEA Nonlinear Differential Equations Appl., 29(1):Paper No. 7, 38, 2022

    work page 2022

  52. [52]

    Gebhard, J

    B. Gebhard, J. J. Kolumb´ an, and L. Sz´ ekelyhidi. A new approach to the Rayleigh-Taylor instability.Arch. Ration. Mech. Anal., 241(3):1243–1280, 2021

    work page 2021

  53. [53]

    $C^{1/5^{-}}$ Convex Integration Solutions of Ideal MHD

    M. Giardi and L. Sz´ ekelyhidi, Jr.C1/5− Convex Integration Solutions of Ideal MHD.arXiv:2604.12091, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  54. [54]

    A. Gilbert. Fast dynamo action in the ponomarenko dynamo.Geophysical & Astrophysical Fluid Dynamics, 44:241–258, 1988

    work page 1988

  55. [55]

    A. D. Gilbert. Dynamo theory. InHandbook of mathematical fluid dynamics, Vol. II, pages 355–441. North- Holland, Amsterdam, 2003

    work page 2003

  56. [56]

    V. Giri, H. Kwon, and M. Novack. TheL 3-based strong Onsager theorem.arXiv:2305.18509, 2023

    work page arXiv 2023

  57. [57]

    Giri and R

    V. Giri and R. a.-O. Radu. The Onsager conjecture in 2D: a Newton-Nash iteration.Invent. Math., 238(2):691– 768, 2024

    work page 2024

  58. [58]

    Hitruhin and S

    L. Hitruhin and S. Lindberg. Relaxation of the kinematic dynamo equations.Proc. Amer. Math. Soc., 152(12):5265–5278, 2024

    work page 2024

  59. [59]

    P. Isett. A proof of Onsager’s conjecture.Ann. of Math. (2), 188(3):871–963, 2018

    work page 2018

  60. [60]

    Kampschulte.Gradient flows and a generalized Wasserstein distance in the space of Cartesian currents

    M. Kampschulte.Gradient flows and a generalized Wasserstein distance in the space of Cartesian currents. PhD Thesis, RWTH Aachen University, 2017

    work page 2017

  61. [61]

    K¨ apyl¨ a

    P. K¨ apyl¨ a. Connecting mean-field theory with dynamo simulations.Living Rev Sol Phys, 22(3):77 pp., 2025

    work page 2025

  62. [62]

    S. G. Krantz and H. R. Parks.Geometric integration theory. Cornerstones. Birkh¨ auser Boston, Inc., Boston, MA, 2008

    work page 2008

  63. [63]

    Krause and K.-H

    F. Krause and K.-H. R¨ adler.Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon Press, 1980

    work page 1980

  64. [64]

    Y. Li, Z. Zeng, and D. Zhang. Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations.J. Math. Pures Appl. (9), 165:232–285, 2022

    work page 2022

  65. [65]

    Markfelder.Convex integration applied to the multi-dimensional compressible Euler equations, volume 2294 of Lecture Notes in Mathematics

    S. Markfelder.Convex integration applied to the multi-dimensional compressible Euler equations, volume 2294 of Lecture Notes in Mathematics. Springer, Cham, [2021]©2021

    work page 2021

  66. [66]

    F. Mengual. H-principle for the 2-dimensional incompressible porous media equation with viscosity jump.Anal. PDE, 15(2):429–476, 2022. TURBULENT DYNAMOS AND THE GEOMETRIC TRANSPORT EQUATION 49

    work page 2022

  67. [67]

    Mengual and L

    F. Mengual and L. Sz´ ekelyhidi, Jr. Dissipative Euler flows for vortex sheet initial data without distinguished sign.Comm. Pure Appl. Math., 76(1):163–221, 2023

    work page 2023

  68. [68]

    C. Miao, Y. Nie, and W. Ye. On Onsager-type conjecture for the Els¨ asser energies of the ideal MHD equations. Ann. PDE, 11(2):Paper No. 31, 77, 2025

    work page 2025

  69. [69]

    Miao and W

    C. Miao and W. Ye. On the weak solutions for the MHD systems with controllable total energy and cross helicity. J. Math. Pures Appl. (9), 181:190–227, 2024

    work page 2024

  70. [70]

    Modena and G

    S. Modena and G. Sattig. Convex integration solutions to the transport equation with full dimensional concen- tration.Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 37(5):1075–1108, 2020

    work page 2020

  71. [71]

    Modena and L

    S. Modena and L. Sz´ ekelyhidi, Jr. Non-uniqueness for the transport equation with Sobolev vector fields.Ann. PDE, 4(2):Paper No. 18, 38, 2018

    work page 2018

  72. [72]

    Moffatt and E

    K. Moffatt and E. Dormy.Self-exciting fluid dynamos. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2019

    work page 2019

  73. [73]

    Navarro-Fern´ andez and D

    V. Navarro-Fern´ andez and D. Villinger. Spectral instability in the smooth Ponomarenko dynamo. arXiv:2509.19201, 2025

    work page arXiv 2025

  74. [74]

    Nu˜ nez and J

    M. Nu˜ nez and J. Sanz. Uniform growth rates for the magnetic field in a kinematic dynamo.Journal of Physics, 33:3605–3611, 2000

    work page 2000

  75. [75]

    Y. B. Ponomarenko. Theory of the hydromagnetic generator.J. Appl. Mech. Tech. Phys., 14:775–778, 1973

    work page 1973

  76. [76]

    F. Rindler. Space-time integral currents of bounded variation.Calc. Var. Partial Differential Equations, 62(2):Pa- per No. 54, 31, 2023

    work page 2023

  77. [77]

    K. Rowan. A subsequentially fast dynamo onT 3.arXiv:2505.23936, 2025

    work page arXiv 2025

  78. [78]

    Sattig and L

    G. Sattig and L. S. Jr. The baire category method for intermittent convex integration.Acta Math. Hungar., 171(1):88–106, 2023

    work page 2023

  79. [79]

    Sorella and D

    M. Sorella and D. Villringer. A limsup fast dynamo onR 3.arXiv:2511.23024, 2025

    work page arXiv 2025

  80. [80]

    A. Soward. Fast dynamo action in a steady flow.J. Fluid Mech., 180:267—-295, 1987

    work page 1987

Showing first 80 references.