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Almost sure global well-posedness for 3D Euler equation and other fluid dynamics models

Juraj Foldes, Mouhamadou Sy

TL;DR

The paper develops a probabilistic framework to obtain almost-sure global well-posedness for 3D Euler and related fluid models by constructing statistical ensembles via fluctuation-dissipation and Galerkin approximations. It proves the existence of invariant measures for finite-dimensional Galerkin truncations, passes to the inviscid limit, and establishes a global flow on a comprehensive statistical ensemble with growth bounds; invariance and moment bounds are established, and, under additional invariants, large-data results and infinite-dimensionality of the support follow. The approach extends to generalized SQG and shell models, demonstrating global trajectories on ensembles even when deterministic global well-posedness is unknown. These results illuminate how stochastic forcing and dissipation, balanced and sent to zero, can select globally regular dynamics in high-dimensional fluid systems and provide a robust probabilistic notion of global behavior. The work has potential implications for understanding long-time behavior, invariant measures, and data-ensemble perspectives in fluid dynamics beyond deterministic theories.

Abstract

We construct various statistical ensembles associated to the 3D Euler equations and prove global regularity of these equations for data living on these sets. Similar results are also proven for generalized SQG equations and some shell models. Qualitative properties of the ensembles and the constructed flows are also given.

Almost sure global well-posedness for 3D Euler equation and other fluid dynamics models

TL;DR

The paper develops a probabilistic framework to obtain almost-sure global well-posedness for 3D Euler and related fluid models by constructing statistical ensembles via fluctuation-dissipation and Galerkin approximations. It proves the existence of invariant measures for finite-dimensional Galerkin truncations, passes to the inviscid limit, and establishes a global flow on a comprehensive statistical ensemble with growth bounds; invariance and moment bounds are established, and, under additional invariants, large-data results and infinite-dimensionality of the support follow. The approach extends to generalized SQG and shell models, demonstrating global trajectories on ensembles even when deterministic global well-posedness is unknown. These results illuminate how stochastic forcing and dissipation, balanced and sent to zero, can select globally regular dynamics in high-dimensional fluid systems and provide a robust probabilistic notion of global behavior. The work has potential implications for understanding long-time behavior, invariant measures, and data-ensemble perspectives in fluid dynamics beyond deterministic theories.

Abstract

We construct various statistical ensembles associated to the 3D Euler equations and prove global regularity of these equations for data living on these sets. Similar results are also proven for generalized SQG equations and some shell models. Qualitative properties of the ensembles and the constructed flows are also given.
Paper Structure (15 sections, 28 theorems, 332 equations)

This paper contains 15 sections, 28 theorems, 332 equations.

Table of Contents

  1. Introduction
  2. Statement of the results.
  3. Notations
  4. Uniform local theory, and local convergence
  5. Fluctuation-Dissipation and Galerkin approximation
  6. Stochastic Global well posedness
  7. Stationary measures and moment bounds
  8. Inviscid limit
  9. Statistical ensemble and global flow
  10. General context
  11. Global well-posedness for 3D Euler
  12. Invariance of the measure
  13. Absolute continuity and large data properties for 3D Euler
  14. Large data
  15. Infinite dimensionality

Key Result

Theorem 1.1

Assume Euler1 preserves the $L^2-$norm, that is, $t \mapsto \|u(t)\|_{L^2}$ is constant in time (see assumptions cnclg--zmcd on $B$ below). Then there exists a set $\Sigma\subset H^s({\mathbb{T}}^d)$ such that If in addition the equation Euler1 possesses another coercive conservation law (other than $L^2$ norm), then the equation admits a non-trivial invariant measure $\mu$ such that $\mu(\Sigma)

Theorems & Definitions (60)

  • Theorem 1.1
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4: Remark on the statistical ensemble $\Sigma^{Euler}$ associated to the $3D$ Euler system
  • Theorem 1.5
  • Proposition 2.1
  • Corollary 2.2
  • ...and 50 more