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Probabilistic Approaches to The Energy Equality in Forced Surface Quasi-Geostrophic Equations

Lin Wang, Zhengyan Wu

TL;DR

The paper develops a probabilistic framework for the forced dissipative surface quasi-geostrophic (SQG) equation on the torus, connecting stochastic large deviations to the deterministic energy equality. It establishes a dynamical large deviation principle restricted to a weak-strong uniqueness class, and shows energy equality holds on time-reversible subsets of the closure; a restricted quasi-potential is analyzed and linked to Gaussian fluctuations under a conditional equivalence. The approach blends exponential-martingale upper bounds, entropy-method lower bounds, and a robust commutator-based treatment of nonlinearities in rough Sobolev spaces, with time-reversal symmetry playing a key role in relating energy conservation to rate-function symmetry. The results illuminate how non-Gaussian features in equilibrium may relate to the (non)uniqueness of the deterministic SQG equation, offering a probabilistic lens on a classical PDE regularity/uniqueness problem and establishing a pathway toward a deeper understanding of energy dissipation, stochastic regularization, and quasi-potential structures in active scalar equations.

Abstract

We explore probabilistic approaches to the deterministic energy equality for the forced Surface Quasi-Geostrophic (SQG) equation on a torus. First, we prove the zero-noise dynamical large deviations for a corresponding stochastic SQG equation, where the lower bound matches the upper bound on a certain closure of the weak-strong uniqueness class for the deterministic forced SQG equation. Furthermore, we show that the energy equality for the deterministic SQG equation holds on arbitrary time-reversible subsets of the domain where we match the upper bound and the lower bound. Conversely, the violation of the deterministic energy equality breaks the lower bound of large deviations. These results extend the existing techniques in Gess, Heydecker, and the second author \cite{arXiv:2311.02223} to generalized Sobolev spaces with negative indices. Finally, we provide an analysis of the restricted quasi-potential and prove a conditional equivalence compared to the rate function of large deviations for the Gaussian distribution. This suggests a potential connection between non-Gaussian large deviations in equilibrium for the stochastic SQG equation and the open problem regarding the uniqueness of the deterministic SQG equation.

Probabilistic Approaches to The Energy Equality in Forced Surface Quasi-Geostrophic Equations

TL;DR

The paper develops a probabilistic framework for the forced dissipative surface quasi-geostrophic (SQG) equation on the torus, connecting stochastic large deviations to the deterministic energy equality. It establishes a dynamical large deviation principle restricted to a weak-strong uniqueness class, and shows energy equality holds on time-reversible subsets of the closure; a restricted quasi-potential is analyzed and linked to Gaussian fluctuations under a conditional equivalence. The approach blends exponential-martingale upper bounds, entropy-method lower bounds, and a robust commutator-based treatment of nonlinearities in rough Sobolev spaces, with time-reversal symmetry playing a key role in relating energy conservation to rate-function symmetry. The results illuminate how non-Gaussian features in equilibrium may relate to the (non)uniqueness of the deterministic SQG equation, offering a probabilistic lens on a classical PDE regularity/uniqueness problem and establishing a pathway toward a deeper understanding of energy dissipation, stochastic regularization, and quasi-potential structures in active scalar equations.

Abstract

We explore probabilistic approaches to the deterministic energy equality for the forced Surface Quasi-Geostrophic (SQG) equation on a torus. First, we prove the zero-noise dynamical large deviations for a corresponding stochastic SQG equation, where the lower bound matches the upper bound on a certain closure of the weak-strong uniqueness class for the deterministic forced SQG equation. Furthermore, we show that the energy equality for the deterministic SQG equation holds on arbitrary time-reversible subsets of the domain where we match the upper bound and the lower bound. Conversely, the violation of the deterministic energy equality breaks the lower bound of large deviations. These results extend the existing techniques in Gess, Heydecker, and the second author \cite{arXiv:2311.02223} to generalized Sobolev spaces with negative indices. Finally, we provide an analysis of the restricted quasi-potential and prove a conditional equivalence compared to the rate function of large deviations for the Gaussian distribution. This suggests a potential connection between non-Gaussian large deviations in equilibrium for the stochastic SQG equation and the open problem regarding the uniqueness of the deterministic SQG equation.

Paper Structure

This paper contains 16 sections, 24 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Key idea and technical comment
  4. Comments of literature
  5. Structure of the paper
  6. Preliminary
  7. Notations and spaces
  8. The Regularization of Noise
  9. Basic results of large deviations
  10. Existence and Exponential Tightness
  11. Characterization of Rate Function
  12. Upper Bound for Large Deviations
  13. Lower Bound for Large Deviations
  14. Probabilistic Approaches to the Energy Equality
  15. Characterization of Quasi-potential
  16. ...and 1 more sections

Key Result

Theorem 1.3

For every $\varepsilon,\delta(\varepsilon)>0$, let $\theta_{\varepsilon,\delta(\varepsilon)}$ be a stochastic generalized Leray solution of SPDE-SQG in the sense of Definition def-stochastic-leray with initial data $\theta_{\varepsilon, \delta(\varepsilon)}(0)$ satisfying Assumption Gaussian-initial and for any open set $G\subset\mathbb{X}_{\alpha,\beta}$,

Figures (1)

  • Figure :

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Proposition \ref{['proposition-upperbound']}, Proposition \ref{['prop-restricted-lower-bound']}
  • Theorem 1.4: Theorem \ref{['theorem-energy-equality']}
  • Corollary 1.5
  • Theorem 1.6: Proposition \ref{['quasi-Gaussian']}
  • Lemma 2.1
  • Lemma 2.2: Commutator Estimate
  • Lemma 2.3
  • Proposition 2.4
  • ...and 36 more