Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative convergence rates for scaling limit of SPDEs with transport noise

Franco Flandoli, Lucio Galeati, Dejun Luo

Abstract

We consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamical equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic arguments, especially heat kernel properties and maximal estimates for stochastic convolutions. Similar ideas are applied to the stochastic 2D Keller-Segel model, yielding explicit choice of noise to ensure that the blow-up probability is less than any given threshold. Our approach also gives rise to some mixing property for stochastic linear transport equations and dissipation enhancement in the viscous case.

Quantitative convergence rates for scaling limit of SPDEs with transport noise

Abstract

We consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamical equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic arguments, especially heat kernel properties and maximal estimates for stochastic convolutions. Similar ideas are applied to the stochastic 2D Keller-Segel model, yielding explicit choice of noise to ensure that the blow-up probability is less than any given threshold. Our approach also gives rise to some mixing property for stochastic linear transport equations and dissipation enhancement in the viscous case.

Paper Structure

This paper contains 21 sections, 21 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Quantitative convergence rate for 2D fluid models
  3. Explicit estimates on blow-up probability
  4. Some results on mixing and dissipation enhancement
  5. Quantitative finite horizon mixing
  6. Dissipation enhancement
  7. Our strategy
  8. Preliminaries
  9. Some analytical lemmas
  10. Maximal estimates on stochastic convolution
  11. Mild formulation of the equation
  12. Proofs of Theorem \ref{['thm-1']} and related models
  13. Proofs of Theorem \ref{['thm-1']} and Corollary \ref{['cor-approx-uniq']}
  14. 2D Boussinesq system
  15. mSQG equations
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

Let $\omega$ and $\tilde{\omega}$ be weak solutions to stoch-NS and determ-NS respectively and assume $\omega$ satisfies solu-bounds. Then, for any $\alpha\in(0,1)$, there exists some $C=C(\alpha)>0$ such that for any $\varepsilon\in(0,\alpha]$, one has

Theorems & Definitions (46)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 36 more