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The primitive equations with rough transport noise: Global well-posedness and regularity

Antonio Agresti

TL;DR

The paper develops global well-posedness and instantaneous regularization results for the primitive equations under rough transport noise with Hölder regularity γ>1/2. By working in an $L^q$ setting with $q>2$ and identifying critical anisotropic Besov spaces $B^{2/q}_{(q,2),p}$ that reflect vertical anisotropy, the authors achieve stochastic maximal regularity-based well-posedness even for Kraichnan-type noise with α>1 and for Kolmogorov-spectrum-type forcing. The framework extends to non-isothermal models and Stratonovich formulations, providing a robust treatment of turbulence-like transport in geophysical flows. These results advance the mathematical understanding of stochastic PDEs in anisotropic, non-Hilbert settings and have potential implications for modeling turbulent oceanic and atmospheric dynamics.

Abstract

In this paper we establish global well-posedness and instantaneous regularization results for the primitive equations with transport noise of Hölder regularity $ γ>\frac{1}{2}$. It is known that if $γ<1$, then the noise is too rough for a strong formulation of primitive equations in an $L^2$-based setting. To handle rough noise, we crucially use $L^q$-techniques with $q> 2$. Interestingly, we identify a family of critical anisotropic Besov spaces for primitive equations, which is new even in the deterministic case. The behavior of these spaces reflects the intrinsic anisotropy of the primitive equations and plays an essential role in establishing global well-posedness and regularization. Our results cover Kraichnan's type noise with correlation greater than one, and as a by-product, a 2D noise reproducing the Kolmogorov spectrum of turbulence. Moreover, the instantaneous regularization is new also in the widely studied case of $H^1$-data and $γ>1 $.

The primitive equations with rough transport noise: Global well-posedness and regularity

TL;DR

The paper develops global well-posedness and instantaneous regularization results for the primitive equations under rough transport noise with Hölder regularity γ>1/2. By working in an setting with and identifying critical anisotropic Besov spaces that reflect vertical anisotropy, the authors achieve stochastic maximal regularity-based well-posedness even for Kraichnan-type noise with α>1 and for Kolmogorov-spectrum-type forcing. The framework extends to non-isothermal models and Stratonovich formulations, providing a robust treatment of turbulence-like transport in geophysical flows. These results advance the mathematical understanding of stochastic PDEs in anisotropic, non-Hilbert settings and have potential implications for modeling turbulent oceanic and atmospheric dynamics.

Abstract

In this paper we establish global well-posedness and instantaneous regularization results for the primitive equations with transport noise of Hölder regularity . It is known that if , then the noise is too rough for a strong formulation of primitive equations in an -based setting. To handle rough noise, we crucially use -techniques with . Interestingly, we identify a family of critical anisotropic Besov spaces for primitive equations, which is new even in the deterministic case. The behavior of these spaces reflects the intrinsic anisotropy of the primitive equations and plays an essential role in establishing global well-posedness and regularization. Our results cover Kraichnan's type noise with correlation greater than one, and as a by-product, a 2D noise reproducing the Kolmogorov spectrum of turbulence. Moreover, the instantaneous regularization is new also in the widely studied case of -data and .
Paper Structure (45 sections, 32 theorems, 243 equations)

This paper contains 45 sections, 32 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Illustration -- Global well-posedness in the Kolmogorov regime
  3. Scaling and criticality
  4. Discussion on non-isothermal models and overview
  5. Comparison with the deterministic setting
  6. Notation
  7. Preliminaries
  8. Smoothness of the Kraichnan model
  9. Hydrostatic Helmholtz projection and related function spaces
  10. Statement of the main results -- The isothermal case
  11. Reformulation of the primitive equations
  12. Main assumption and $(p,\kappa,\delta,q)$-solutions
  13. Local well-posedness and regularity in critical spaces
  14. Serrin's type blow-up criteria and high-order regularity
  15. Global well-posedness
  16. ...and 30 more sections

Key Result

Theorem 1.1

Suppose that and that there exists $r\in (0,2)$ such that a.e. on ${\mathbb R}_+\times \Omega\times \mathbb{T}^d$ and for all $\xi\in {\mathbb R}^3$ Then for all $v_0\in H^1(\mathbb{T}^3;{\mathbb R}^2)$, $\theta_0\in H^1(\mathbb{T}^3)$ such that $\int_{\mathbb{T}_z} \nabla_{x,y}\cdot v_0(\cdot,z)\,\mathrm{d} z=0$, the stochastic PEs eq:primitive_intro admits a unique global solution which instan

Theorems & Definitions (66)

  • Theorem 1.1: Global well-posedness of PEs with rough transport noise -- Informal version
  • Proposition 2.1
  • proof
  • Example 3.2: Kolmogorov's spectrum of turbulence via 2d transport noise
  • Remark 3.3: Weakening the assumptions on $f,g$
  • Definition 3.4: $(p,\kappa,\delta,q)$-solution
  • Theorem 3.5: Local existence and uniqueness in critical spaces
  • Proposition 3.6: Local continuity
  • Theorem 3.7: Instantaneous regularization I
  • Corollary 3.8: Compatibility
  • ...and 56 more