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

Slim Ibrahim, Quyuan Lin, Lingjun Qian, Edriss S. Titi

TL;DR

The work analyzes finite-time singularity formation in the hydrostatic (primitive) equations with either non-diffusive or diffusive temperature dynamics. By reducing the 3D system to a 1D trace model on the vertical axis and employing a self-similar bootstrap framework, the authors obtain precise blowup profiles with $a(t,Z) \sim (T-t)^{-1}$ near $Z=0$ and characterize the temperature perturbations $c(t,z)$ under both diffusion settings. The core finding is that temperature variation, whether transported or diffused, does not alter the blowup mechanism or the stability of the velocity singularity, though the diffusive case requires refined weighted-energy estimates. The results are connected back to the original coordinates, establishing stability under small analytic perturbations and providing a rigorous link between the self-similar framework and physical space. These insights clarify the robustness of singularity formation in geophysical flows under non-constant temperature coupling.

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.

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

TL;DR

The work analyzes finite-time singularity formation in the hydrostatic (primitive) equations with either non-diffusive or diffusive temperature dynamics. By reducing the 3D system to a 1D trace model on the vertical axis and employing a self-similar bootstrap framework, the authors obtain precise blowup profiles with near and characterize the temperature perturbations under both diffusion settings. The core finding is that temperature variation, whether transported or diffused, does not alter the blowup mechanism or the stability of the velocity singularity, though the diffusive case requires refined weighted-energy estimates. The results are connected back to the original coordinates, establishing stability under small analytic perturbations and providing a rigorous link between the self-similar framework and physical space. These insights clarify the robustness of singularity formation in geophysical flows under non-constant temperature coupling.

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.

Paper Structure

This paper contains 24 sections, 21 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Derivation and Reduction
  3. Temporal and Spatial Rescaling-Self-similar Ansatz
  4. System of Equations and Vanishing-Speed Condition at Origin for Non-diffusive Case ($\sigma=0$)
  5. System of Equations and Vanishing-Speed condition at Origin for Diffusive Case ($\sigma=1$)
  6. More Invariant Vanishing Speed at Origin
  7. Bootstrap Argument for the case of Non-Diffusive Temperature
  8. Definitions
  9. A-priori Estimates
  10. Interior Estimates
  11. Solution to the Interior Differential Inequalities-Choice of Parameters
  12. Exterior Estimates
  13. Conclusion
  14. Bootstrap Argument for the case of Diffusive Temperature
  15. Definitions
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\phi(x)=\exp(-x)$. There exists $\lambda^{*}_0>0$, sufficiently small, such that for all $0<\lambda_0\leq \frac{\lambda_0^*}{2}$, there exists $\kappa(\lambda_0)>0$ such that the following holds. If the initial conditions to System trace-system-introduction are of the form of satisfying the zero-average condition and spatial scaling $\nu_0$ are in the range of: and the perturbations $\tilde

Theorems & Definitions (51)

  • Theorem 1: Non-diffusive Temperature ($\sigma=0$)
  • Remark 1
  • Theorem 2: Diffusive Temperature ($\sigma=1$)
  • Remark 2
  • Corollary 1: Invariance of Vanishing Speed at Zero
  • proof
  • Remark 3
  • Definition 1: Initial closeness
  • Definition 2: Trapped on $\left(s_0,s_1\right)$
  • Remark 4
  • ...and 41 more