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arxiv: 2605.29866 · v1 · pith:3LGSGRZWnew · submitted 2026-05-28 · 🧮 math.AP

Vorticity blow-up for the 2D incompressible non-homogeneous Euler equations with uniform C^{1,sqrt{frac{4}{3}}-1-varepsilon} force

Diego C\'ordoba , Andr\'es La\'in-Sanclemente , Luis Mart\'inez-Zoroa This is my paper

Pith reviewed 2026-06-29 06:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords non-homogeneous Euler equationsvorticity blow-upfinite-time singularityHölder continuous forceincompressible fluid dynamicsBoussinesq approximation
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The pith

Solutions of the 2D non-homogeneous Euler equations can develop finite-time singularities with a force of limited spatial regularity.

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

This paper establishes that solutions to the 2D incompressible non-homogeneous Euler equations can develop singularities in finite time. The force term is allowed to be continuous in time, Hölder continuous in space with exponent just below √(4/3)-1, and square-integrable in space. The result is obtained by adapting a blow-up construction previously used for the Boussinesq system. If correct, this indicates that the threshold for regularity of the force to guarantee smooth solutions is at least as low as this Hölder class.

Core claim

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with C^0_t C^{1,√(4/3)-1-ε}_x ∩ C^0_t L^2_x source terms that develop a singularity in finite time by adapting the Boussinesq blow-up construction.

What carries the argument

Adaptation of the Boussinesq blow-up construction to the non-homogeneous Euler equations, which preserves the finite-time singularity.

Load-bearing premise

The Boussinesq blow-up construction can be adapted to the non-homogeneous Euler equations without losing the finite-time singularity or the required regularity of the force.

What would settle it

Computation of the adapted solution fields to check if they satisfy the non-homogeneous Euler equations with the given force class and exhibit the predicted vorticity blow-up at a finite time.

read the original abstract

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C^{0}_{t}C^{1,\,\sqrt{\frac{4}{3}}-1-\varepsilon}_{x}\cap C^{0}_{t}L^{2}_{x}$ source terms that develop a singularity in finite time. In order to achieve this, we adapt the Boussinesq blow-up we set up in arXiv:2505.20988 to the non-homogeneous Euler setting. Furthermore, we bring the potential existence of two different types of singularities of the forced system to light.

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 / 2 minor

Summary. The paper claims to establish the existence of solutions to the 2D incompressible non-homogeneous Euler equations featuring source terms in the space C^0_t C^{1,√(4/3)-1-ε}_x ∩ C^0_t L^2_x that develop a finite-time singularity in the vorticity. The construction is obtained by adapting the Boussinesq blow-up previously constructed in arXiv:2505.20988, and the work additionally indicates the possible existence of two distinct singularity types for the forced system.

Significance. If the adaptation of the prior Boussinesq construction succeeds while preserving all required estimates, the result would supply a concrete example of finite-time vorticity blow-up for the non-homogeneous Euler system under a force of limited Hölder regularity. This would extend known singularity-formation results to a new equation and could illuminate mechanisms for two different blow-up scenarios. The manuscript's strength lies in its explicit attempt to transfer an existing construction rather than starting from scratch.

major comments (1)
  1. [Adaptation section (likely §3 or §4)] The central existence claim rests on the successful adaptation of the Boussinesq construction from arXiv:2505.20988. The manuscript asserts that this adaptation preserves both the finite-time singularity and the stated regularity of the force, yet provides no explicit verification or re-derivation of the key estimates (e.g., those controlling the force term in C^{1,√(4/3)-1-ε} and L^2) under the new transport and pressure terms of the non-homogeneous Euler system. This verification is load-bearing for the main theorem.
minor comments (2)
  1. The abstract refers to 'two different types of singularities' but the main text should explicitly define and distinguish these types with reference to the constructed solutions.
  2. Notation for the force regularity class should be introduced once and used consistently; the intersection C^0_t C^{1,√(4/3)-1-ε}_x ∩ C^0_t L^2_x appears without prior definition in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comment point by point below.

read point-by-point responses

  1. Referee: The central existence claim rests on the successful adaptation of the Boussinesq construction from arXiv:2505.20988. The manuscript asserts that this adaptation preserves both the finite-time singularity and the stated regularity of the force, yet provides no explicit verification or re-derivation of the key estimates (e.g., those controlling the force term in C^{1,√(4/3)-1-ε} and L^2) under the new transport and pressure terms of the non-homogeneous Euler system. This verification is load-bearing for the main theorem.

    Authors: We agree that the manuscript would benefit from a more explicit verification of the estimates under the adapted system. In the revised version we will add a dedicated subsection that re-derives the bounds on the force term in C^{1,√(4/3)-1-ε} ∩ L^2, carefully accounting for the differences in the transport and pressure terms between the Boussinesq and non-homogeneous Euler equations. This will confirm preservation of the required regularity and support the finite-time singularity. revision: yes

Circularity Check

1 steps flagged

Central blow-up construction adapted from overlapping-author prior work

specific steps
  1. self citation load bearing [Abstract]
    "In order to achieve this, we adapt the Boussinesq blow-up we set up in arXiv:2505.20988 to the non-homogeneous Euler setting."

    The finite-time vorticity blow-up (the central claim) is justified solely by reference to a prior construction whose authors overlap with the present paper; the adaptation step is asserted but the load-bearing singularity mechanism reduces to the cited self-work.

full rationale

The paper's existence result for finite-time singularity in the non-homogeneous Euler system is obtained by adapting the Boussinesq construction from arXiv:2505.20988 (overlapping authors). This self-citation is load-bearing for the singularity itself, though the adaptation to a new equation and force regularity supplies partial independent content. No internal self-definitional loops or fitted-input predictions are present in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of the adaptation from the cited Boussinesq construction together with standard local existence theory for the Euler equations in the given function spaces; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The Boussinesq blow-up construction from arXiv:2505.20988 remains valid after the indicated modifications to the non-homogeneous Euler system.
    The abstract states that the result is obtained by adapting that construction.
  • standard math Standard local well-posedness and continuation criteria apply to the non-homogeneous Euler equations with the given force class.
    Implicit in any finite-time blow-up existence argument for these PDEs.

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Reference graph

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