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arxiv: 2510.10090 · v2 · submitted 2025-10-11 · 🧮 math.AP

On the Profile of Singularity Formation for the Incompressible Hydrostatic Boussinesq system

Slim Ibrahim , Quyuan Lin , Lingjun Qian , Edriss S. Titi This is my paper

Pith reviewed 2026-05-18 08:03 UTC · model grok-4.3

classification 🧮 math.AP
keywords singularity formationhydrostatic Boussinesq systemprimitive equationsstabilitytemperature variationfinite time blowupincompressible fluids
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The pith

Variation of temperature does not affect singularity formation or stability in the velocity field of the hydrostatic Boussinesq system.

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

This paper examines the incompressible hydrostatic Boussinesq system, which models large-scale ocean and atmosphere dynamics. It establishes that non-constant temperature, whether without diffusion or with vertical diffusivity added, leaves the finite-time stable singularity in the velocity field unchanged from the known isothermal case. A sympathetic reader would care because the result shows the velocity singularity behaves robustly under temperature coupling, so models can retain the same singular profiles without temperature altering the core blow-up mechanism.

Core claim

The authors show that the singularity profile previously found for the velocity in the constant-temperature hydrostatic Euler equations persists as a stable attractor when the system is coupled to a non-constant temperature equation. This stability holds in both the case with no temperature diffusion and the case with vertical diffusivity in the temperature dynamics. As a result, temperature variations neither trigger nor prevent the velocity singularity formation.

What carries the argument

Stability of the velocity singularity profile as an attractor under the coupled temperature dynamics, which carries the argument by showing the profile remains unchanged.

Load-bearing premise

The initial data and temperature profiles are chosen so that the velocity singularity profile from the constant-temperature case persists as a stable attractor under the coupled temperature dynamics.

What would settle it

A numerical simulation of the full system from the paper's chosen initial data that develops a velocity singularity at a different time or with a different profile when temperature varies would disprove the stability claim.

read the original abstract

The primitive equations (PEs) model planetary large-scale oceanic and atmospheric dynamics. While it has been shown that there are smooth solutions to the inviscid PEs (also called the hydrostatic Euler equations) with constant temperature (isothermal) that develop stable singularities in finite time, the effect of non-constant temperature on the singularity formation has not been established yet. This paper studies the stability of singularity formation for non-constant temperature in two scenarios: when there is no diffusion in the temperature, or when a vertical diffusivity is added to the temperature dynamics. For both scenarios, our results indicate that the variation of temperature affects neither the formation of singularity, nor its stability, in the velocity field, respectively.

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 studies singularity formation in the incompressible hydrostatic Boussinesq system (primitive equations) with non-constant temperature. It establishes that, for both the pure transport case and the case with vertical temperature diffusivity, the velocity field develops the same stable finite-time singularity as in the previously studied isothermal setting; temperature variations neither prevent blow-up nor destabilize the velocity profile.

Significance. If the estimates close without hidden restrictions, the result shows that the hydrostatic singularity mechanism is robust to temperature coupling. This strengthens the applicability of the isothermal singularity constructions to more realistic Boussinesq models of large-scale ocean/atmosphere flow and clarifies the subcritical role of buoyancy forcing relative to the velocity blow-up.

major comments (2)
  1. [Theorem 1.1 / stability analysis] The central stability claim requires that the buoyancy term remains a controlled perturbation of the known isothermal velocity profile. The manuscript should explicitly state (in the main theorem or in the a-priori estimates section) whether this control holds for arbitrary smooth initial temperature or only under a smallness condition on the initial temperature deviation; the abstract and introduction give no indication of such a restriction.
  2. [Section 3 (transport case)] In the transport case, the temperature equation is a linear transport by the singular velocity field. The proof must verify that the resulting temperature gradients do not feed back through the hydrostatic pressure to alter the vertical velocity or the blow-up time; a concrete estimate showing that the temperature contribution to the pressure gradient remains o(1) relative to the leading singular term is needed.
minor comments (2)
  1. [Introduction and preliminaries] Notation for the hydrostatic pressure and the buoyancy term should be introduced once and used consistently; several places appear to switch between p and the integrated temperature without comment.
  2. [Section 2] The comparison with the isothermal results of the self-cited papers would benefit from a short table or paragraph listing the precise differences in the estimates that arise from the temperature coupling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment in turn below, providing clarifications on the scope of the results and indicating where revisions will be made to improve explicitness.

read point-by-point responses

  1. Referee: [Theorem 1.1 / stability analysis] The central stability claim requires that the buoyancy term remains a controlled perturbation of the known isothermal velocity profile. The manuscript should explicitly state (in the main theorem or in the a-priori estimates section) whether this control holds for arbitrary smooth initial temperature or only under a smallness condition on the initial temperature deviation; the abstract and introduction give no indication of such a restriction.

    Authors: The estimates in the a-priori section (Section 2) and the stability analysis for Theorem 1.1 are constructed to control the buoyancy term as a perturbation for arbitrary smooth initial temperature data, without any smallness assumption on the initial deviation. The hydrostatic balance and the structure of the singular velocity profile allow the temperature contribution to be absorbed into the existing bounds independently of its size. We will revise the statement of Theorem 1.1 and add a clarifying sentence in the introduction to make this explicit. revision: yes

  2. Referee: [Section 3 (transport case)] In the transport case, the temperature equation is a linear transport by the singular velocity field. The proof must verify that the resulting temperature gradients do not feed back through the hydrostatic pressure to alter the vertical velocity or the blow-up time; a concrete estimate showing that the temperature contribution to the pressure gradient remains o(1) relative to the leading singular term is needed.

    Authors: In the transport case the temperature is passively advected, and the hydrostatic pressure is recovered by vertical integration. The resulting contribution to the vertical velocity equation is estimated in the proof of the main result in Section 3; because the temperature remains bounded while the leading singular term in the velocity blows up, the temperature-induced term is indeed o(1) relative to the principal singular contribution as the blow-up time is approached. This control is used to show that neither the vertical velocity nor the blow-up time is altered. To make the comparison more transparent we will insert a short remark immediately after the relevant estimate that isolates the o(1) statement. revision: partial

Circularity Check

0 steps flagged

Minor self-citation for background; central stability result independent

full rationale

The paper extends prior results on finite-time singularity formation in the isothermal hydrostatic Euler equations to the non-isothermal Boussinesq case by establishing that the velocity blow-up profile persists under temperature perturbations (both transport and vertically diffusive). The derivation proceeds via a priori estimates and perturbation analysis around the known singular solution, with initial data and temperature profiles selected to ensure the buoyancy term remains subcritical. This selection is part of the theorem hypothesis rather than a fitted input renamed as prediction. Self-citations appear for the base isothermal singularity but are not load-bearing for the new stability claim, which rests on fresh estimates for the coupled system. No step reduces the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result relies on the existence of a stable singular profile from the constant-temperature case and on the assumption that temperature perturbations remain controlled by the velocity blow-up. No new free parameters are introduced in the abstract; the analysis inherits the function-space setting and initial-data assumptions from prior isothermal work.

axioms (1)
  • domain assumption The singular velocity profile constructed for the isothermal inviscid primitive equations remains an attractor when temperature is allowed to vary.
    This is the load-bearing premise that allows the stability statement to carry over to the Boussinesq system.

pith-pipeline@v0.9.0 · 5655 in / 1222 out tokens · 29023 ms · 2026-05-18T08:03:54.322042+00:00 · methodology

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Pascal Az´ erad and Francisco Guill´ en. Mathematical justification of the hydrostatic approximation in the primi- tive equations of geophysical fluid dynamics.SIAM Journal on Mathematical Analysis, 33(4):847–859, January 2001

    work page 2001

  2. [2]

    Ill-posedness of the hydrostatic euler-boussinesq equations and failure of hydrostatic limit, 2024

    Roberta Bianchini, Michele Coti Zelati, and Lucas Ertzbischoff. Ill-posedness of the hydrostatic euler-boussinesq equations and failure of hydrostatic limit, 2024

    work page 2024

  3. [3]

    Chongsheng Cao, Slim Ibrahim, Kenji Nakanishi, and Edriss S. Titi. Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics.Communications in Mathematical Physics, 337(2):473–482, April 2015

    work page 2015

  4. [4]

    Chongsheng Cao, Jinkai Li, and Edriss S. Titi. Global well-posedness of the 3d primitive equations with hori- zontal viscosity and vertical diffusivity.Physica D: Nonlinear Phenomena, 412:132606, November 2020

    work page 2020

  5. [5]

    Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics.Annals of Mathematics, 166(1):245–267, July 2007

    Chongsheng Cao and Edriss Titi. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics.Annals of Mathematics, 166(1):245–267, July 2007

    work page 2007

  6. [6]

    Stable singularity formation for the inviscid primitive equations

    Charles Collot, Slim Ibrahim, and Quyuan Lin. Stable singularity formation for the inviscid primitive equations. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 41(2):317–356, June 2023

    work page 2023

  7. [7]

    Note on loss of regularity for solutions of the 3?d incompressible euler and related equations

    Petre Constantin. Note on loss of regularity for solutions of the 3?d incompressible euler and related equations. Communications in Mathematical Physics, 104(2):311–326, June 1986

    work page 1986

  8. [8]

    Daniel Han-Kwan and Toan T. Nguyen. Ill-posedness of the hydrostatic euler and singular vlasov equations. Archive for Rational Mechanics and Analysis, 221(3):1317–1344, February 2016

    work page 2016

  9. [9]

    Global strong well-posedness of the three dimensional primitive equations inL p-spaces.Archive for Rational Mechanics and Analysis, 221(3):1077–1115, February 2016

    Matthias Hieber and Takahito Kashiwabara. Global strong well-posedness of the three dimensional primitive equations inL p-spaces.Archive for Rational Mechanics and Analysis, 221(3):1077–1115, February 2016

    work page 2016

  10. [10]

    Louis N. Howard. Note on a paper of john w. miles.Journal of Fluid Mechanics, 10(04):509, June 1961

    work page 1961

  11. [11]

    Slim Ibrahim, Quyuan Lin, and Edriss S. Titi. Finite-time blowup and ill-posedness in sobolev spaces of the inviscid primitive equations with rotation.Journal of Differential Equations, 286:557–577, June 2021

    work page 2021

  12. [12]

    Loss of regularity for the 2d euler equations.Journal of Mathematical Fluid Mechanics, 23(4), September 2021

    In-Jee Jeong. Loss of regularity for the 2d euler equations.Journal of Mathematical Fluid Mechanics, 23(4), September 2021

    work page 2021

  13. [13]

    Vicol, and Mohammed Ziane

    Igor Kukavica, Roger Temam, Vlad C. Vicol, and Mohammed Ziane. Local existence and uniqueness for the hydrostatic euler equations on a bounded domain.Journal of Differential Equations, 250(3):1719–1746, February 2011

    work page 2011

  14. [14]

    On the regularity of the primitive equations of the ocean.Nonlinearity, 20(12):2739–2753, October 2007

    Igor Kukavica and Mohammed Ziane. On the regularity of the primitive equations of the ocean.Nonlinearity, 20(12):2739–2753, October 2007

    work page 2007

  15. [15]

    Jinkai Li and Edriss S. Titi. The primitive equations as the small aspect ratio limit of the navier–stokes equa- tions: Rigorous justification of the hydrostatic approximation.Journal de Math´ ematiques Pures et Appliqu´ ees, 124:30–58, April 2019

    work page 2019

  16. [16]

    Titi, and Guozhi Yuan

    Jinkai Li, Edriss S. Titi, and Guozhi Yuan. The primitive equations approximation of the anisotropic horizontally viscous 3d navier-stokes equations.Journal of Differential Equations, 306:492–524, January 2022

    work page 2022

  17. [17]

    New formulations of the primitive equations of atmosphere and applications

    J L Lions, R Temam, and S Wang. New formulations of the primitive equations of atmosphere and applications. Nonlinearity, 5(2):237–288, March 1992

    work page 1992

  18. [18]

    Xin Liu and Edriss S. Titi. Rigorous justification of the hydrostatic approximation limit of viscous compressible flows.Physica D: Nonlinear Phenomena, 464:134195, August 2024

    work page 2024

  19. [19]

    John W. Miles. On the stability of heterogeneous shear flows.Journal of Fluid Mechanics, 10(04):496, June 1961

    work page 1961

  20. [20]

    Ill-posedness of the hydrostatic euler and navier–stokes equations.Archive for Rational Me- chanics and Analysis, 194(3):877–886, January 2009

    Michael Renardy. Ill-posedness of the hydrostatic euler and navier–stokes equations.Archive for Rational Me- chanics and Analysis, 194(3):877–886, January 2009

    work page 2009

  21. [21]

    PhD thesis, Lulea Uni- versity of Technology, 2003

    Anna Wedesting.Weighted Inequalities of Hardy-Type and their Limiting Inequalities. PhD thesis, Lulea Uni- versity of Technology, 2003

    work page 2003

  22. [22]

    Blowup of solutions of the hydrostatic euler equations.Proceedings of the American Mathe- matical Society, 143(3):1119–1125, November 2014

    Tak Kwong Wong. Blowup of solutions of the hydrostatic euler equations.Proceedings of the American Mathe- matical Society, 143(3):1119–1125, November 2014

    work page 2014