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Strong Uniqueness by Kraichnan Transport Noise for the 2D Boussinesq Equations with Zero Viscosity

Shuaijie Jiao, Dejun Luo

TL;DR

The paper addresses the well-posedness of the inviscid 2D Boussinesq equations under Kraichnan-type transport noise, establishing existence and pathwise uniqueness of probabilistic strong solutions for initial vorticity in $L^p$ and initial temperature in $L^2$ by leveraging a noise-induced anomalous regularity in $H^{- rac{α}{}}$. The approach builds smooth approximate solutions with mollified data and noise, plus a small viscosity, and proves convergence in $C_T\dot{H}^{-1}$ using stopping times to obtain $L^p$ control of the vorticity; the key mechanism is the negative-definite contribution in the Itô-formula energy estimate that results in an $H^{-α}$ regularity gain due to Kraichnan noise. The main contributions are (i) extending pathwise uniqueness by incorporating anomalous regularity to the coupled vorticity–temperature system, (ii) establishing uniform energy bounds and almost sure convergence for a range of $p$ and $α$, and (iii) providing a robust framework for convergence from regularized to true solutions, including a treatment of the endpoint regime via decomposition. The results demonstrate that transport-type stochastic forcing can regularize a challenging fluid-thermodynamic system and yield strong well-posedness results in a probabilistic sense, with potential implications for stochastic fluid models exhibiting turbulence-like noise structures.

Abstract

We investigate the inviscid 2D Boussinesq equations driven by rough transport noise of Kraichnan type with regularity index $α\in (0,1/2)$. For all $1<p<\infty$, we establish the existence and uniqueness of probabilistic strong solutions for all $L^p$ initial vorticity and $L^2$ initial temperature, under the parameter constraint $0<α< 1-1/(p\wedge 2)$. The key ingredient is the anomalous regularity due to the noise proven by Coghi and Maurelli \cite{CogMau} who dealt with stochastic 2D Euler equations. Combining techniques from analysis and probability, we demonstrate how the additional regularity from noise compensates the singularity due to the nonlinear parts and coupled terms.

Strong Uniqueness by Kraichnan Transport Noise for the 2D Boussinesq Equations with Zero Viscosity

TL;DR

The paper addresses the well-posedness of the inviscid 2D Boussinesq equations under Kraichnan-type transport noise, establishing existence and pathwise uniqueness of probabilistic strong solutions for initial vorticity in and initial temperature in by leveraging a noise-induced anomalous regularity in . The approach builds smooth approximate solutions with mollified data and noise, plus a small viscosity, and proves convergence in using stopping times to obtain control of the vorticity; the key mechanism is the negative-definite contribution in the Itô-formula energy estimate that results in an regularity gain due to Kraichnan noise. The main contributions are (i) extending pathwise uniqueness by incorporating anomalous regularity to the coupled vorticity–temperature system, (ii) establishing uniform energy bounds and almost sure convergence for a range of and , and (iii) providing a robust framework for convergence from regularized to true solutions, including a treatment of the endpoint regime via decomposition. The results demonstrate that transport-type stochastic forcing can regularize a challenging fluid-thermodynamic system and yield strong well-posedness results in a probabilistic sense, with potential implications for stochastic fluid models exhibiting turbulence-like noise structures.

Abstract

We investigate the inviscid 2D Boussinesq equations driven by rough transport noise of Kraichnan type with regularity index . For all , we establish the existence and uniqueness of probabilistic strong solutions for all initial vorticity and initial temperature, under the parameter constraint . The key ingredient is the anomalous regularity due to the noise proven by Coghi and Maurelli \cite{CogMau} who dealt with stochastic 2D Euler equations. Combining techniques from analysis and probability, we demonstrate how the additional regularity from noise compensates the singularity due to the nonlinear parts and coupled terms.
Paper Structure (12 sections, 11 theorems, 189 equations)

This paper contains 12 sections, 11 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main results
  4. Main strategy
  5. Structure of paper and notations
  6. Approximate solutions
  7. Construction of approximate solutions
  8. Uniform energy estimate
  9. Almost sure convergence in $C_T \dot{H}^{-1}$
  10. Proof of Theorem \ref{['main-thm']}
  11. Well-posedness of the regularized model \ref{['regular-model']}
  12. Proof of Corollary \ref{['cor3']}

Key Result

Theorem 1.1

Given $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P}, (W^k)_k )$, where $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ is a filtered probability space satisfying the usual conditions and $(W^k)_k$ are a sequence of independent $(\mathcal{F}_t)_t$-adapted real Brownian motions. Let $1<p and the following equalities hold in distribution sense for every $t\in [0,T]$,

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Remark 2.5
  • ...and 15 more