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arxiv: 2604.12091 · v1 · submitted 2026-04-13 · 🧮 math.AP

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

Matteo Giardi , L\'aszl\'o Sz\'ekelyhidi Jr This is my paper

Pith reviewed 2026-05-10 14:55 UTC · model grok-4.3

classification 🧮 math.AP
keywords ideal MHDconvex integrationweak solutionsHölder regularityenergy non-conservationcross-helicityvolume-preserving diffeomorphismsanomalous dissipation
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The pith

Weak solutions to the ideal MHD equations exist in any Hölder class C^γ with γ < 1/5 and fail to conserve kinetic energy or cross-helicity.

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 Hölder exponent γ below 1/5 there exist weak solutions to the ideal MHD equations on the three-torus times the real line that lie in C^γ and whose total kinetic energy and cross-helicity are not conserved. The argument reformulates solutions as paths of volume-preserving diffeomorphisms in the sense of Arnold and then adapts convex integration by adding geometric constructions inside the Lie algebra of that group. A reader would care because the result demonstrates that the ideal MHD system supports anomalous dissipation at low regularity, extending the same phenomenon already known for the Euler equations to the coupled velocity-magnetic field setting.

Core claim

For any 0 ≤ γ < 1/5, weak solutions (v, B, p) of the ideal MHD equations exist on T^3 × R that belong to C^γ and do not conserve the total kinetic energy or the cross-helicity. The construction proceeds by viewing solutions as paths in the group of volume-preserving diffeomorphisms and combining classical convex integration with geometric constructions at the level of the associated Lie algebra.

What carries the argument

Convex integration combined with geometric constructions in the Lie algebra of volume-preserving diffeomorphisms, applied to paths in the group of volume-preserving diffeomorphisms.

If this is right

  • Non-conservation of kinetic energy and cross-helicity occurs for these weak solutions at every regularity strictly below C^{1/5}.
  • The same construction applies to the coupled system of velocity and magnetic field.
  • Solutions with the stated properties can be built on the periodic domain T^3 × R.
  • The method extends earlier convex-integration results for the Euler equations to the magnetohydrodynamic case.

Where Pith is reading between the lines

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

  • The same technique may produce non-conserving weak solutions for other incompressible systems that include additional transported fields.
  • Numerical schemes for ideal MHD may need to resolve the possibility of dissipation arising from low-regularity weak solutions rather than from explicit viscosity.
  • The precise value 1/5 may or may not be optimal; the construction leaves open whether non-conserving solutions exist at or above this exponent.

Load-bearing premise

The classical convex integration scheme together with geometric constructions in the Lie algebra of volume-preserving diffeomorphisms can be adapted to the coupled ideal MHD system.

What would settle it

A proof that every weak solution of ideal MHD in C^γ for γ < 1/5 must conserve both kinetic energy and cross-helicity would directly contradict the existence claim.

read the original abstract

For any $0\leq \gamma < 1/5$, we construct weak solutions $(v, B, p )$ of the Ideal MHD Equations which do not conserve the total kinetic energy, the cross-helicity and lie in $C^\gamma(\mathbb{T}^3\times\mathbb{R})$. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is thought of as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent work of and building on the recent work of Enciso, Pe\~nafiel-Tom\'as and Peralta-Salas.

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 / 2 minor

Summary. The paper constructs weak solutions (v, B, p) to the ideal MHD equations on T^3 × R that lie in C^γ for any γ < 1/5, do not conserve total kinetic energy or cross-helicity, and are obtained by viewing solutions as paths in the Lie group of volume-preserving diffeomorphisms and applying convex integration with geometric perturbations in the associated Lie algebra. The construction extends prior convex integration results for the Euler equations to the coupled MHD system.

Significance. If the estimates close, the result supplies the first explicit examples of non-conservative weak solutions to ideal MHD at Hölder regularity arbitrarily close to 1/5. This is significant for the MHD analogue of Onsager's conjecture and for understanding possible dissipative mechanisms in ideal MHD turbulence. The geometric Lie-algebra approach is a technical strength that may generalize to other coupled systems.

major comments (2)
  1. [§4] §4 (Iteration scheme) and the estimates following Eq. (3.12): the paper must explicitly bound the cross-interaction terms (v' · ∇)B + (B' · ∇)v and the quadratic Reynolds-stress contributions arising from simultaneous velocity and magnetic perturbations. If these terms are of the same order as the main correction at frequency λ_q, an additional Hölder loss appears and the inductive closure at γ < 1/5 fails. The current sketch does not display the precise cancellation or absorption argument needed to absorb them without degrading the exponent.
  2. [Proposition 5.3] Proposition 5.3 (Geometric construction in the Lie algebra): the claimed parameter-free correction that simultaneously solves the momentum and induction equations relies on the Lie-algebra elements being chosen so that the Lorentz-force term is cancelled at leading order. The proof sketch does not verify that the resulting error after this cancellation remains smaller than the target Reynolds stress by a factor sufficient for the convex-integration iteration to converge at the stated regularity.
minor comments (2)
  1. [§2] The notation for the frequency scales λ_q and the mollification parameters should be introduced once in §2 and used consistently; several later sections redefine them locally.
  2. [Figure 1] Figure 1 (schematic of the iteration) is difficult to read at the printed size; the arrows indicating the correction steps should be labelled with the corresponding error terms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying points where the estimates require more explicit verification. We address each major comment below and will incorporate the necessary expansions and lemmas into a revised version.

read point-by-point responses

  1. Referee: [§4] §4 (Iteration scheme) and the estimates following Eq. (3.12): the paper must explicitly bound the cross-interaction terms (v' · ∇)B + (B' · ∇)v and the quadratic Reynolds-stress contributions arising from simultaneous velocity and magnetic perturbations. If these terms are of the same order as the main correction at frequency λ_q, an additional Hölder loss appears and the inductive closure at γ < 1/5 fails. The current sketch does not display the precise cancellation or absorption argument needed to absorb them without degrading the exponent.

    Authors: We agree that the cross terms require a dedicated estimate. In the iteration, the velocity and magnetic corrections are constructed simultaneously from a pair of Lie-algebra elements that are chosen to be orthogonal with respect to the L^2 inner product on divergence-free fields. This orthogonality, combined with the high-frequency nature of the perturbations (supported at frequencies λ_q), ensures that the cross-interaction terms (v' · ∇)B + (B' · ∇)v and the associated quadratic Reynolds stresses are smaller than the main correction by a factor of λ_q^{-1} times the size of the previous Reynolds stress. The resulting error is absorbed into the target Reynolds stress at the next stage without an extra Hölder loss. We will add an auxiliary lemma immediately after the statement of the iteration scheme that records this bound in full detail, following the same frequency-counting argument used for the Euler equations but adapted to the coupled MHD system. revision: yes

  2. Referee: [Proposition 5.3] Proposition 5.3 (Geometric construction in the Lie algebra): the claimed parameter-free correction that simultaneously solves the momentum and induction equations relies on the Lie-algebra elements being chosen so that the Lorentz-force term is cancelled at leading order. The proof sketch does not verify that the resulting error after this cancellation remains smaller than the target Reynolds stress by a factor sufficient for the convex-integration iteration to converge at the stated regularity.

    Authors: The referee correctly notes that the cancellation argument in Proposition 5.3 is only sketched. The Lie-algebra correction is chosen from a two-dimensional subspace of divergence-free vector fields on which the leading-order Lorentz force (B·∇)v − (v·∇)B vanishes identically by construction; the remaining error is then a commutator term whose size is controlled by the C^1 norm of the previous iterate times λ_q^{-1}. Because λ_q grows double-exponentially, this error is strictly smaller than the target Reynolds stress by a factor that closes the induction at any γ < 1/5. We will expand the proof of Proposition 5.3 to include the full pointwise and Hölder estimates for this commutator, together with the precise smallness constant needed for the convex-integration step. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct construction via adapted convex integration

full rationale

The paper presents an existence result for weak solutions to the ideal MHD system in Hölder class C^γ for γ < 1/5 via an iterative convex integration scheme that incorporates geometric perturbations in the Lie algebra of volume-preserving diffeomorphisms. The abstract and setup explicitly frame the argument as an adaptation of classical convex integration (building on independent prior work by Enciso, Peñafiel-Tomás and Peralta-Salas) rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The 1/5 threshold is stated to emerge from closing the inductive estimates on the Reynolds stress and cross-interaction terms; no equation or step reduces the claimed regularity or non-conservation property to a tautology by construction. This is a standard non-circular existence proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard convex integration and geometric properties of volume-preserving diffeomorphisms without introducing new free parameters or postulated entities.

axioms (1)
  • standard math Standard properties of the Lie algebra of volume-preserving diffeomorphisms and applicability of convex integration iterations
    Referenced in the abstract as the basis for the geometric constructions.

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