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arxiv: 2604.14100 · v1 · submitted 2026-04-15 · 🧮 math.AP

The 2D Euler equations are well-posed for generic initial data in L²

Lucio Galeati This is my paper

Pith reviewed 2026-05-10 12:07 UTC · model grok-4.3

classification 🧮 math.AP
keywords 2D Euler equationsweak solutionsBaire categoryresidual setwell-posednessNavier-Stokesvanishing viscosityLagrangian flow
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The pith

A residual set of L² initial data makes the 2D Euler equations globally well-posed.

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

The paper shows that a comeager collection of divergence-free initial data in L² supports unique global weak solutions to the incompressible 2D Euler equations on the plane or the torus. These solutions conserve kinetic energy, appear as the zero-viscosity limit of Navier-Stokes solutions, and drive a unique regular Lagrangian flow that makes the transport equation well-posed. On the torus the solutions also arise as Galerkin limits. This generic well-posedness is obtained by combining the known global existence of smooth solutions with weak-strong uniqueness in a Baire-category argument.

Core claim

There exists a residual set, in the Baire sense, of divergence-free initial data u₀ in L²(D), where D is either the plane or the torus, such that the 2D Euler equations admit global unique weak solutions. These solutions satisfy the energy balance, are obtained as the vanishing viscosity limit from 2D Navier-Stokes, possess a unique regular Lagrangian flow, make the transport equation well-posed, and on the torus are limits of Galerkin approximations. The proof relies on global existence of smooth solutions and weak-strong uniqueness.

What carries the argument

Baire category theorem applied to the comeager set of initial data for which weak-strong uniqueness holds with respect to smooth solutions.

If this is right

  • Solutions conserve kinetic energy exactly.
  • Anomalous dissipation is absent in the vanishing viscosity limit.
  • A unique regular Lagrangian flow exists for the velocity.
  • The linear transport equation is well-posed.
  • Galerkin approximations converge on the torus.

Where Pith is reading between the lines

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

  • Ill-posedness phenomena must occur on a meager set of initial data.
  • Baire-category methods could extend to other equations with global smooth solutions.
  • Generic data may allow reliable numerical approximation without anomalous behavior.

Load-bearing premise

Weak-strong uniqueness must hold on a comeager set of initial data in L², which in turn requires that smooth solutions are sufficiently dense.

What would settle it

Finding a specific non-meager collection of initial data in L² where either multiple weak solutions exist or energy is not conserved would disprove the residual-set claim.

read the original abstract

In this note we show the existence of a residual set (in the sense of Baire) of divergence free initial data $u_0\in L^2(D)$, $D=\mathbb{R}^2$ or $\mathbb{T}^2$, for which global existence and uniqueness of weak solutions to the incompressible 2D Euler equations holds. The associated solutions $u$ satisfy the energy balance and are recovered in the vanishing viscosity limit from solutions to 2D Navier-Stokes, which as a consequence cannot display anomalous dissipation of energy. Additionally, there exists a unique regular Lagrangian flow associated to such $u$, and the associated transport equation is well-posed. Finally, when $D=\mathbb{T}^2$, the solution $u$ is recovered as the limit of Galerkin approximations. The proof relies on global existence of smooth solutions and weak-strong uniqueness arguments.

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 / 1 minor

Summary. The manuscript claims that there exists a residual (comeager) set of divergence-free initial data u₀ ∈ L²(D) (D = ℝ² or 𝕋²) for which the incompressible 2D Euler equations admit a unique global weak solution u. This solution satisfies the energy balance, is recovered as the vanishing-viscosity limit of 2D Navier-Stokes solutions, possesses a unique regular Lagrangian flow, and (when D = 𝕋²) is the limit of Galerkin approximations. The argument invokes the known global existence of smooth solutions together with weak-strong uniqueness to construct the residual set via the Baire category theorem.

Significance. If the density of the good set is established, the result would be significant: it identifies a comeager subset of L² initial data on which the 2D Euler equations are well-posed in the weak sense, despite the existence of non-unique weak solutions for some data. The additional properties (energy equality, vanishing-viscosity recovery, unique Lagrangian flow) rule out anomalous dissipation on this generic set and connect the Euler problem to Navier-Stokes and transport theory. The approach of using Baire category to capture generic well-posedness is a recognized technique in the field.

major comments (1)
  1. [Proof section (following the abstract)] The central construction of the residual set via Baire category requires that the set of 'good' initial data (those for which weak-strong uniqueness holds with the smooth solution) be dense in L². While smooth divergence-free data are dense, the manuscript does not supply an explicit argument showing that every L²-ball contains initial data for which any weak solution must coincide with the unique smooth solution obtained by approximation. Without a density proof, the good set need not be comeager and the residual-set claim fails. This is load-bearing for the main theorem.
minor comments (1)
  1. [Abstract] The abstract states that the proof 'relies on global existence of smooth solutions and weak-strong uniqueness arguments' but does not indicate where in the manuscript the density step is carried out; a one-sentence roadmap would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for a more explicit density argument in the Baire-category construction. We agree that this step is load-bearing and will be clarified in the revision.

read point-by-point responses

  1. Referee: [Proof section (following the abstract)] The central construction of the residual set via Baire category requires that the set of 'good' initial data (those for which weak-strong uniqueness holds with the smooth solution) be dense in L². While smooth divergence-free data are dense, the manuscript does not supply an explicit argument showing that every L²-ball contains initial data for which any weak solution must coincide with the unique smooth solution obtained by approximation. Without a density proof, the good set need not be comeager and the residual-set claim fails. This is load-bearing for the main theorem.

    Authors: We agree that an explicit verification of density is required and was not spelled out in the original manuscript. In the revised version we will insert a short, self-contained paragraph immediately after the definition of the good set. The argument is as follows: the collection of smooth, divergence-free vector fields is dense in L²(D). For any such smooth initial datum v₀ the corresponding smooth solution of the Euler equations is the unique weak solution (by the classical weak-strong uniqueness theorem). Consequently every smooth initial datum lies in the good set. Since the smooth data are dense, the good set itself is dense in L². This density, together with the G_δ property already established in the manuscript, yields that the good set is comeager, completing the application of the Baire category theorem. We will also add a brief remark clarifying that the same density holds on both ℝ² and 𝕋². revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim rests on independent classical results

full rationale

The paper establishes a residual set of L² initial data for which 2D Euler weak solutions exist, are unique, satisfy energy balance, and arise as vanishing-viscosity limits. Its proof strategy explicitly invokes two external, previously established facts: global existence of smooth solutions to 2D Euler/Navier-Stokes and weak-strong uniqueness. These are classical results (Yudovich-type regularity and associated uniqueness theorems) that predate the present work and are not derived from any construction, fit, or self-citation internal to the paper. No equation or definition in the abstract reduces the target residual set to a tautology or to a parameter fitted from the same data; the Baire-category argument is applied to a set whose density and comeager character are asserted on the basis of those independent inputs. Consequently the derivation chain does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results in PDE theory rather than new free parameters or invented entities.

axioms (2)
  • standard math Global existence of smooth solutions to the 2D incompressible Euler and Navier-Stokes equations
    Invoked explicitly in the abstract as the foundation for the weak-strong uniqueness argument.
  • standard math Weak-strong uniqueness principle for 2D Euler weak solutions
    Used to transfer uniqueness from smooth to generic data; cited as part of the proof strategy.

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Forward citations

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