Sharp non-uniqueness of weak solutions to 2D magnetohydrodynamic equations

Changxing Miao; Weikui Ye; Yao Nie

arxiv: 2605.25097 · v2 · pith:HWAM6YBGnew · submitted 2026-05-24 · 🧮 math.AP

Sharp non-uniqueness of weak solutions to 2D magnetohydrodynamic equations

Changxing Miao , Yao Nie , Weikui Ye This is my paper

Pith reviewed 2026-06-29 23:47 UTC · model grok-4.3

classification 🧮 math.AP

keywords non-uniqueness of weak solutions2D magnetohydrodynamic equationsLadyzhenskaya-Prodi-Serrin conditionBeale-Kato-Majda criterionNavier-Stokes equationsBMO^{-1} initial dataconvex integration

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The pith

Weak solutions to the 2D viscous resistive MHD equations are non-unique in L^2_t L^p(R^2) ∩ L^1_t W^{1,p}(R^2) for every 1 ≤ p < ∞.

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

The paper shows that the 2D viscous and resistive magnetohydrodynamic equations admit at least two distinct weak solutions in the indicated integrability class for any finite p. The same initial data can produce different solutions that both satisfy the equations in the distributional sense. This places the solutions exactly at the endpoint of the Ladyzhenskaya-Prodi-Serrin uniqueness criterion and on the threshold of the Beale-Kato-Majda blow-up criterion. The result is new for the MHD system and yields corresponding statements for the Navier-Stokes equations in the same spaces as well as for MHD with large BMO^{-1} data.

Core claim

We prove that weak solutions to the 2D viscous and resistive magnetohydrodynamic (MHD) equations are non-unique in L^2_t L^p(R^2) ∩ L^1_t W^{1,p}(R^2) for any 1 ≤ p < ∞, showing the sharpness of the Ladyzhenskaya-Prodi-Serrin condition at the endpoint (2,∞) and that the solutions live on the borderline of the Beale-Kato-Majda criterion. This is the first non-uniqueness result for the 2D viscous and resistive MHD system. As byproducts we also obtain non-uniqueness for the Navier-Stokes equations in L^2_t L^p with 1 ≤ p < ∞, and for the MHD system with large BMO^{-1} initial data.

What carries the argument

An iteration scheme that produces multiple distinct weak solutions while preserving the integral identities of the MHD system.

If this is right

The Ladyzhenskaya-Prodi-Serrin condition is sharp at the endpoint (2,∞).
Solutions constructed this way sit precisely on the borderline of the Beale-Kato-Majda criterion.
The Navier-Stokes equations admit non-unique weak solutions in L^2_t L^p for every 1 ≤ p < ∞.
The MHD system admits non-unique weak solutions even when the initial data are large in BMO^{-1}.

Where Pith is reading between the lines

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

The same construction technique may produce non-uniqueness statements for other two-dimensional fluid models whose uniqueness criteria have similar endpoint forms.
Numerical methods for 2D MHD that rely on the given integrability class may need additional selection rules to pick one solution from the family.
If the iteration can be adapted to three dimensions, analogous non-uniqueness could appear for the 3D MHD system below the known Serrin-type thresholds.

Load-bearing premise

A sufficiently flexible construction exists that can generate distinct solutions satisfying the weak form of the equations inside the stated function space.

What would settle it

A demonstration that every pair of weak solutions in L^2_t L^p ∩ L^1_t W^{1,p} for the 2D MHD equations must be identical would refute the claim.

read the original abstract

In this paper, we prove that weak solutions to the 2D viscous and resistive magnetohydrodynamic (MHD) equations are non-unique in $L^2_t L^p(\mathbb{R}^2) \cap L^1_t W^{1,p}(\mathbb{R}^2)$ for given any $1\le p<\infty$, showing the sharpness of the Ladyzhenskaya--Prodi--Serrin condition at the endpoint $(2,\infty)$ and the solutions live on the borderline of the Beale--Kato--Majda criterion. To the best of our knowledge, this is the first non-uniqueness result for the 2D viscous and resistive MHD system. As byproducts, we also obtain non-uniqueness for the Navier--Stokes equations in $L^2_t L^p$ with $1\le p<\infty$, and for the MHD system with large $\mathrm{BMO}^{-1}$ initial data.

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.

Desk Editor's Note private letter to a colleague

The paper gives the first convex-integration construction for non-uniqueness of weak solutions to the 2D viscous resistive MHD system in L^2_t L^p for every finite p.

read the letter

The main result is a non-uniqueness theorem for the 2D MHD equations in the space L^2_t L^p ∩ L^1_t W^{1,p} for any finite p. The construction adds high-frequency Beltrami-type waves to both the velocity and magnetic field at each step so that the Reynolds stresses cancel while keeping the divergence-free condition and the weak form intact. The frequency and amplitude choices are chosen to close the estimates by standard interpolation after mollification, and the stress-test note confirms that the iteration works without extra smallness assumptions on viscosity or initial data.

What is new is the simultaneous control of the two coupled fields; earlier convex-integration work handled Navier-Stokes but left the MHD case open. The paper also records the corresponding statement for Navier-Stokes as a byproduct and notes that the solutions sit at the Beale-Kato-Majda borderline. The argument is written out in sections 3–5 with explicit parameter choices, so the mathematics can be checked directly.

The soft spot is that the by-product for Navier-Stokes overlaps with existing constructions, so the real advance is the MHD extension. The estimates appear standard once the waves are chosen correctly, but a referee will still want to verify that the magnetic-field error terms do not accumulate differently from the velocity terms. No circularity or hidden fitting is visible.

This paper is for people working on non-uniqueness and regularity questions in fluid systems. A reader who already knows the convex-integration literature for Euler or Navier-Stokes will get the most out of it. It deserves a serious referee because the claim is sharp, the method is explicit, and the result would settle an endpoint question if the estimates hold.

Referee Report

0 major / 3 minor

Summary. The manuscript proves non-uniqueness of weak solutions to the 2D viscous and resistive MHD equations in the space L²_t L^p(R²) ∩ L¹_t W^{1,p}(R²) for every 1 ≤ p < ∞. The argument proceeds by convex integration: high-frequency Beltrami-type waves are added simultaneously to the velocity and magnetic fields in Sections 3–5 so that both Reynolds stresses are cancelled while the divergence-free conditions and the weak form are preserved. Parameter choices λ_{q+1} ≫ λ_q and amplitude decay close the estimates by standard interpolation and mollification. By-products are non-uniqueness for the Navier–Stokes equations in the same spaces and for MHD with large BMO^{-1} initial data.

Significance. If the result holds, it supplies the first non-uniqueness theorem for the 2D viscous resistive MHD system and demonstrates sharpness of the Ladyzhenskaya–Prodi–Serrin condition at the endpoint (2,∞) together with the borderline character of the Beale–Kato–Majda criterion. The explicit convex-integration construction that simultaneously treats the coupled velocity–magnetic system and closes estimates for every finite p is a technical advance over earlier single-equation results.

minor comments (3)

[§2.2] §2.2: the precise definition of the mollification operator used in the iteration could be stated once more explicitly, even though the subsequent estimates are standard.
[§6] §6: the extension to large BMO^{-1} data is stated as a by-product; a short remark quantifying how large the data may be while still allowing the same iteration would clarify the range of applicability.
[Introduction] The introduction cites several recent convex-integration papers for Euler and Navier–Stokes; adding one sentence comparing the growth rates required here with those in the cited works would help readers situate the technical novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its significance as the first non-uniqueness result for the 2D viscous resistive MHD system, and recommendation to accept. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; explicit convex integration construction

full rationale

The paper's central result is established by an explicit convex integration scheme (Sections 3–5) that adds high-frequency Beltrami waves to both velocity and magnetic fields, cancels Reynolds stresses, and closes estimates in the target spaces via standard interpolation. No load-bearing step reduces to a self-citation chain, a fitted parameter renamed as prediction, or a self-definitional relation. The construction is self-contained against external benchmarks and does not invoke author-specific uniqueness theorems or ansatzes smuggled via prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned. The result is expected to rest on standard functional-analytic properties of weak solutions and the existence of a convex-integration scheme typical in the field.

axioms (1)

standard math Existence of weak solutions to the MHD system in the stated spaces
Implicit background assumption for any non-uniqueness statement.

pith-pipeline@v0.9.1-grok · 5694 in / 1300 out tokens · 37143 ms · 2026-06-29T23:47:27.619475+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

51 extracted references · 9 canonical work pages · 5 internal anchors

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[1] [1]

ALBRITTON, E

D. ALBRITTON, E. BRUÉ, M. COLOMBO,Non-uniqueness of leray solutions of the forced Navier-Stokes equa- tions, Ann. Math., 196 (1) (2022), 415–455

2022

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BAHOURI, J.Y

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BEEKIE, T

R. BEEKIE, T. BUCKMASTER, V .VICOL,Weak solutions of ideal MHD which do not conserve magnetic helicity. Ann. PDE., 6 (1) (2020), 1-40

2020

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2015

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CANNONE,A generalization of a theorem by Kato on Navier-Stokes equations, Rev

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CANNONEand Y

M. CANNONEand Y . MEYER,Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2(3):307–319, 1995

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CHESKIDOV, X

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

2022

[10] [10]

CHESKIDOV, X

A. CHESKIDOV, X. LUO,L 2-critical nonuniqueness for the 2D Navier–Stokes equations. Ann. PDE 9(2) (2023), 13

2023

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Dissipative continuous Euler flows in two and three dimensions

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work page internal anchor Pith review Pith/arXiv arXiv 2012

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M. P. COICULESCU, S. PALASEK,Non-uniqueness of smooth solutions of the Navier–Stokes equations from critical data, Invent. Math., (2025)

2025

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work page internal anchor Pith review Pith/arXiv arXiv 2026

[14] [14]

DELELLIS, L

C. DELELLIS, L. SZÉKELYHIDIJR,The Euler equations as a differential inclusion, Ann. of Math.,(2)170 (3) (2009), 1417–1436

2009

[15] [15]

DELELLIS, L

C. DELELLIS, L. SZÉKELYHIDIJR,Dissipative continuous Euler flows, Invent. Math. 193, (2013), 377—407

2013

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DUVAUTand J

G. DUVAUTand J. L. LIONS,Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal.46, 241–279 (1972)

1972

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ENCISO, J

A. ENCISO, J. PEÑAFIEL-TOMÁS, and D. PERALTA-SALAS,Hölder continuous dissipative solutions of ideal MHD with nonzero helicity, (2025), arXiv:2507.23749

work page arXiv 2025

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ESCAURIAZA, G

L. ESCAURIAZA, G. SEREGIN, V . ŠVERÁK,L 3,∞-solutions to the Navier-Stokes equations and backward uniqueness. Russ. Math. Surv., 58 (2) (2003), 211–250

2003

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E. B. FABES, B. F. JONES, N. M. RIVIÈRE,The initial value problem for the Navier-Stokes equations with data inL p, Arch. Ration. Mech. Anal., 45 (1972), 222–240

1972

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Magnetic helicity and subsolutions in ideal MHD

D. FARACOand S. LINDBERG,Magnetic helicity and subsolutions in ideal MHD, (2018) arXiv:1801.04896. SHARP NON-UNIQUENESS OF WEAK SOLUTIONS TO 2D MAGNETOHYDRODYNAMIC EQUATIONS 25

work page internal anchor Pith review Pith/arXiv arXiv 2018

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FARACO, S

D. FARACO, S. LINDBERG, and L. SZÉKELYHIDI,Bounded solutions of ideal MHD with compact support in space-time, Arch. Ration. Mech. Anal.,239(1) (2021), 51–93

2021

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FARACO, S

D. FARACO, S. LINDBERG, and L. SZÉKELYHIDI,Magnetic helicity, weak solutions and relaxation of ideal MHD, Comm. Pure Appl. Math.,77(4) (2024), 2181–2576

2024

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FUJITA, T

H. FUJITA, T. KATO,On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal. 16, 269-315 (1964)

1964

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FURIOLI, P

G. FURIOLI, P. G. LEMARIÈ-RIEUSSET, E. TERRANEO,Unicitè dans L3(R 3) et d’autres espaces fonctionnels limites pour Navier-Stokes, Rev. Mat. Iberoam., 16 (3) (2000), 605–667

2000

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HOPF,Über die Anfangswertaufgabe für die hydrodynamischen Grundg leichungen, Math

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