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arxiv: 2606.08924 · v2 · pith:JTDVRZLZnew · submitted 2026-06-08 · 🧮 math.AP

Strong Ill-posedness of the 2d Incompressible Euler Equation in Critical Besov Spaces

Karim R. Shikh Khalil This is my paper

Pith reviewed 2026-07-01 08:04 UTC · model grok-4.3

classification 🧮 math.AP
keywords incompressible Euler equationsstrong ill-posednesscritical Besov spaces2D fluid dynamicsvelocity regularitywell-posedness failure
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The pith

The 2D incompressible Euler equation is strongly ill-posed in the critical Besov spaces B^1_{∞,q} for 1<q<∞.

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

This paper proves that the two-dimensional incompressible Euler equation exhibits strong ill-posedness when the velocity field belongs to the critical Besov space B^1_{∞,q} where q is between 1 and infinity. Strong ill-posedness here refers to the failure of solutions to depend continuously on their initial data in these function spaces. These spaces sit at the scaling-critical level for the Euler equations, meaning they are the lowest regularity where one might hope for a well-posed theory. If true, this implies that mathematical models of ideal fluid flow break down at this regularity threshold, limiting the applicability of existence theorems in borderline spaces.

Core claim

The paper claims that the 2d incompressible Euler Equation is strongly ill-posed for velocity fields in B^1_{∞,q} for 1<q<∞. This is established through arguments showing that the solution map is not continuous at the origin or that solutions do not exist in these spaces, using the specific properties of the Besov norms and the nonlinear structure of the Euler equations.

What carries the argument

The notion of strong ill-posedness in Besov spaces, which uses the failure of continuous dependence on initial data, supported by instability constructions that take advantage of the critical scaling of B^1_{∞,q}.

Load-bearing premise

The technical definition of strong ill-posedness as failure of continuous dependence holds due to the embedding and scaling properties of B^1_{∞,q} that permit construction of unstable sequences.

What would settle it

Constructing a continuous solution map from initial data in B^1_{∞,q} to solutions in the same space for some q in (1,∞) would falsify the result.

Figures

Figures reproduced from arXiv: 2606.08924 by Karim R. Shikh Khalil.

Figure 1
Figure 1. Figure 1: Sketch of initial data and the ill-posedness mechanism [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We prove strong ill-posedness of the 2d incompressible Euler Equation for velocity field in the critical Besov Spaces $B^{1}_{\infty, q}$ for $1<q<\infty$.

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

1 major / 0 minor

Summary. The manuscript claims to prove strong ill-posedness of the 2D incompressible Euler equation for velocity fields belonging to the critical Besov spaces B^1_{∞,q} with 1 < q < ∞.

Significance. If the result holds, it would contribute to the literature on the borderline regularity for the Euler equations by establishing ill-posedness in a family of critical Besov spaces, helping to clarify the transition between well-posed and ill-posed regimes.

major comments (1)
  1. The manuscript consists solely of the one-sentence claim in the abstract. No definition of strong ill-posedness is supplied, no counter-example is constructed, and no estimates or functional-analytic arguments are given. Without these elements the central claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments. We agree that the current manuscript is limited to the abstract claim and requires substantial expansion.

read point-by-point responses

  1. Referee: The manuscript consists solely of the one-sentence claim in the abstract. No definition of strong ill-posedness is supplied, no counter-example is constructed, and no estimates or functional-analytic arguments are given. Without these elements the central claim cannot be assessed.

    Authors: We fully agree with the referee's assessment. The submitted manuscript contains only the abstract statement and omits all technical content. We will revise the manuscript to include a precise definition of strong ill-posedness for the 2D Euler equation in B^1_{∞,q}, an explicit construction of initial data demonstrating the failure of continuous dependence, and the supporting estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a direct proof of strong ill-posedness for the 2D Euler equation in B^1_{∞,q} spaces (1<q<∞) with no equations, parameter fits, self-citations, or ansatzes visible in the abstract or context. No load-bearing step reduces to its own inputs by construction, and the result type aligns with standard external literature on critical-space ill-posedness without internal self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; ledger is empty by necessity.

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