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Refined uniqueness results for 2D Euler and gSQG with rough Kraichnan noise

Marco Bagnara, Lucio Galeati

TL;DR

The paper addresses the global well-posedness of stochastic 2D Euler and gSQG equations perturbed by rough Kraichnan-type transport noise in $\mathbb{R}^2$, proving strong existence and pathwise uniqueness for initial data in $L^p$ near the critical scaling. A central innovation is rewriting the nonlinear term to distribute derivatives more effectively, enabling sharp coercivity from the noise in negative Sobolev spaces and overcoming the previous restriction $\alpha<\tfrac{1}{2}$. The authors establish strong existence via vanishing viscosity and then prove pathwise uniqueness for all $\alpha\in(0,1)$ under natural scaling constraints, with a refinement to initial data in the critical class $L^{p_*}$ and, for gSQG, a decomposition requirement $\mathcal{U}^{p_*}$. This work highlights robustness of regularization-by-noise phenomena for turbulent-like active scalars, including the physically relevant case $\alpha=\tfrac{2}{3}$, and provides a rigorous foundation for modeling unresolved scales via rough transport noise. The results have potential implications for understanding anomalous regularization and dissipative mechanisms in stochastic fluid dynamics and may inform related turbulence models and numerical schemes.

Abstract

We prove strong well-posedness results for the stochastic 2D Euler equations in vorticity form and generalized SQG equations, with $L^p$ initial data and driven by a spatially rough, incompressible transport noise of Kraichnan type. Previous works addressed this problem with noise of spatial regularity $α\in (0,1/2)$, in a setting where a rougher noise yields a stronger regularization. We remove this limitation by allowing any $α\in (0,1)$, covering the same range of parameters for which anomalous regularization effects are known to occur in passive scalars. In particular, this covers the physically relevant case $α=2/3$, associated with the Richardson-Kolmogorov scaling of energy cascade.

Refined uniqueness results for 2D Euler and gSQG with rough Kraichnan noise

TL;DR

The paper addresses the global well-posedness of stochastic 2D Euler and gSQG equations perturbed by rough Kraichnan-type transport noise in , proving strong existence and pathwise uniqueness for initial data in near the critical scaling. A central innovation is rewriting the nonlinear term to distribute derivatives more effectively, enabling sharp coercivity from the noise in negative Sobolev spaces and overcoming the previous restriction . The authors establish strong existence via vanishing viscosity and then prove pathwise uniqueness for all under natural scaling constraints, with a refinement to initial data in the critical class and, for gSQG, a decomposition requirement . This work highlights robustness of regularization-by-noise phenomena for turbulent-like active scalars, including the physically relevant case , and provides a rigorous foundation for modeling unresolved scales via rough transport noise. The results have potential implications for understanding anomalous regularization and dissipative mechanisms in stochastic fluid dynamics and may inform related turbulence models and numerical schemes.

Abstract

We prove strong well-posedness results for the stochastic 2D Euler equations in vorticity form and generalized SQG equations, with initial data and driven by a spatially rough, incompressible transport noise of Kraichnan type. Previous works addressed this problem with noise of spatial regularity , in a setting where a rougher noise yields a stronger regularization. We remove this limitation by allowing any , covering the same range of parameters for which anomalous regularization effects are known to occur in passive scalars. In particular, this covers the physically relevant case , associated with the Richardson-Kolmogorov scaling of energy cascade.
Paper Structure (12 sections, 8 theorems, 98 equations)

This paper contains 12 sections, 8 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Main results and comments
  3. Main ideas of proof
  4. Structure of the paper
  5. Preliminaries
  6. Notations and conventions
  7. SPDE in Itô form
  8. Estimates for the noise covariance
  9. Functional estimates for the nonlinearity
  10. Strong existence via vanishing viscosity approximations
  11. Pathwise uniqueness
  12. Technical lemmas

Key Result

Theorem 1.1

Let $\alpha \in (0,1)$, $\beta =0$, $p\in (1,\infty)$ satisfy Then, for any $\theta_0\in L^p \cap \dot H^{-1}$, strong existence and pathwise uniqueness hold for eq:SPDEs_Strat, in the sense of def:solution, in the class

Theorems & Definitions (27)

  • Theorem 1.1: 2D Euler
  • Definition 1.2
  • Theorem 1.3: gSQG
  • Remark 1.4: The space $\mathcal{U}^{p_\star}$
  • Remark 1.5: Further classes of initial data
  • Remark 1.6: The presence of forcing
  • Remark 1.7: Scaling criticality
  • Remark 1.8: Comparison with previous works
  • Remark 1.9: Numerologies and phenomenologies of turbulence
  • Definition 2.1
  • ...and 17 more