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On the local well-posedness of fractionally dissipated primitive equations with transport noise

Ruimeng Hu, Quyuan Lin, Rongchang Liu

TL;DR

This work analyzes the 3D primitive equations with fractional dissipation and Stratonovich transport noise on $\mathbb{T}^3$, addressing local well-posedness in Sobolev spaces. The authors develop novel commutator estimates for the hydrostatic Leray projection to counterbalance the partial dissipation and noisy forcing, and they prove local existence and uniqueness of pathwise solutions in $\mathbb{H}^{\sigma}$ for $\sigma>3$ in the subcritical regime $s\in(1,2)$, with arbitrary data, and in the critical regime $s=1$ under smallness conditions. The proof combines Galerkin approximations, uniform energy bounds, compactness to obtain martingale solutions, and a double-cutoff technique to achieve pathwise uniqueness, yielding a maximal local solution. These results advance the mathematical understanding of stochastic geophysical flows with nonlocal dissipation and transport noise, highlighting the delicate balance between fractional dissipation and noise in 3D.

Abstract

We investigate the three-dimensional fractionally dissipated primitive equations with transport noise, focusing on subcritical and critical dissipation regimes characterized by $ (-Δ)^{s/2} $ with $ s \in (1,2)$ and $s = 1$, respectively. For $σ>3$, we establish the local existence of unique pathwise solutions in Sobolev space $H^σ$. This result applies to arbitrary initial data in the subcritical case ($s \in(1,2)$), and to small initial data in the critical case ($s=1$). The analysis is particularly challenging due to the loss of horizontal derivatives in the nonlinear terms and the lack of full dissipation. To address these challenges, we develop novel commutator estimates involving the hydrostatic Leray projection.

On the local well-posedness of fractionally dissipated primitive equations with transport noise

TL;DR

This work analyzes the 3D primitive equations with fractional dissipation and Stratonovich transport noise on , addressing local well-posedness in Sobolev spaces. The authors develop novel commutator estimates for the hydrostatic Leray projection to counterbalance the partial dissipation and noisy forcing, and they prove local existence and uniqueness of pathwise solutions in for in the subcritical regime , with arbitrary data, and in the critical regime under smallness conditions. The proof combines Galerkin approximations, uniform energy bounds, compactness to obtain martingale solutions, and a double-cutoff technique to achieve pathwise uniqueness, yielding a maximal local solution. These results advance the mathematical understanding of stochastic geophysical flows with nonlocal dissipation and transport noise, highlighting the delicate balance between fractional dissipation and noise in 3D.

Abstract

We investigate the three-dimensional fractionally dissipated primitive equations with transport noise, focusing on subcritical and critical dissipation regimes characterized by with and , respectively. For , we establish the local existence of unique pathwise solutions in Sobolev space . This result applies to arbitrary initial data in the subcritical case (), and to small initial data in the critical case (). The analysis is particularly challenging due to the loss of horizontal derivatives in the nonlinear terms and the lack of full dissipation. To address these challenges, we develop novel commutator estimates involving the hydrostatic Leray projection.
Paper Structure (13 sections, 11 theorems, 154 equations)

This paper contains 13 sections, 11 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the main Result
  3. Functional settings
  4. Assumptions on the noise
  5. Notion of solutions
  6. Main result
  7. Proof of the main result
  8. Analysis of the Galerkin system
  9. Compactness and martingale solutions
  10. Pathwise uniqueness and local pathwise solutions
  11. Remarks on the supercritical case
  12. Auxiliary lemmas
  13. Commutator estimates

Key Result

Theorem 2.3

Let $\sigma>3$, $s\in(1,2)$ and assume the noise coefficient satisfies e.121003. Then, for any $V_0\in L^2(\Omega,\mathbb{H}^{\sigma})$, there exists a maximal pathwise solution $(V,(\eta_n)_{n\geq1},\xi)$ to system e.052502. The same result holds for the case $s=1$ if we assume additionally that th

Theorems & Definitions (22)

  • Definition 2.1: Pathwise solution
  • Definition 2.2: Martingale solution
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • proof : Proof of Theorem \ref{['t.062501']}
  • ...and 12 more