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Continuous data assimilation for 2D stochastic Navier-Stokes equations

Hakima Bessaih, Benedetta Ferrario, Oussama Landoulsi, Margherita Zanella

TL;DR

This work extends the Azouani–Olson–Titi continuous data assimilation framework to 2D stochastic Navier–Stokes equations with additive and multiplicative noise. It derives rigorous convergence results for the nudged solution to the true stochastic flow, with rates and conditions that depend on the noise structure, the nudging strength, and the observation scale. In particular, exponential convergence in expectation is established for additive noise and for multiplicative noise under bounded covariance, while multiplicative noise with sublinear or linear growth yields polynomial rates; pathwise results are provided in the additive case and partially in the multiplicative case. The analysis connects stochastic data assimilation with Foias–Prodi estimates and invariant-measure techniques, delivering a stochastic generalization of AOT with explicit dependence on noise intensity and system parameters.

Abstract

Continuous data assimilation methods, such as the nudging algorithm introduced by Azouani, Olson, and Titi (AOT) [2], are known to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with observed dynamics. In this work, we extend this framework to a stochastic regime by considering the two-dimensional incompressible Navier-Stokes equations subject to either additive or multiplicative noise. We establish sufficient conditions on the nudging parameter and the spatial observation scale that guarantee convergence of the nudged solution to the true stochastic flow. In the case of multiplicative noise, convergence holds in expectation, with exponential or polynomial rates depending on the growth of the noise covariance. For additive noise, we obtain the exponential convergence both in expectation and pathwise. These results yield a stochastic generalization of the AOT theory, demonstrating how the interplay between random forcing, viscous dissipation and feedback control governs synchronization in stochastic fluid systems.

Continuous data assimilation for 2D stochastic Navier-Stokes equations

TL;DR

This work extends the Azouani–Olson–Titi continuous data assimilation framework to 2D stochastic Navier–Stokes equations with additive and multiplicative noise. It derives rigorous convergence results for the nudged solution to the true stochastic flow, with rates and conditions that depend on the noise structure, the nudging strength, and the observation scale. In particular, exponential convergence in expectation is established for additive noise and for multiplicative noise under bounded covariance, while multiplicative noise with sublinear or linear growth yields polynomial rates; pathwise results are provided in the additive case and partially in the multiplicative case. The analysis connects stochastic data assimilation with Foias–Prodi estimates and invariant-measure techniques, delivering a stochastic generalization of AOT with explicit dependence on noise intensity and system parameters.

Abstract

Continuous data assimilation methods, such as the nudging algorithm introduced by Azouani, Olson, and Titi (AOT) [2], are known to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with observed dynamics. In this work, we extend this framework to a stochastic regime by considering the two-dimensional incompressible Navier-Stokes equations subject to either additive or multiplicative noise. We establish sufficient conditions on the nudging parameter and the spatial observation scale that guarantee convergence of the nudged solution to the true stochastic flow. In the case of multiplicative noise, convergence holds in expectation, with exponential or polynomial rates depending on the growth of the noise covariance. For additive noise, we obtain the exponential convergence both in expectation and pathwise. These results yield a stochastic generalization of the AOT theory, demonstrating how the interplay between random forcing, viscous dissipation and feedback control governs synchronization in stochastic fluid systems.

Paper Structure

This paper contains 24 sections, 22 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Summary of contributions, challenges and main results
  3. Plan of the paper
  4. Mathematical setting
  5. Notations
  6. Functional spaces and operators
  7. Assumptions on the stochastic term
  8. The multiplicative noise case
  9. The additive noise case
  10. Interpolant operators
  11. Data assimilation in the deterministic framework
  12. Data assimilation in the stochastic framework: the multiplicative noise case
  13. Convergence in expectation
  14. Some partial results for pathwise convergence
  15. Data assimilation in the stochastic framework: the additive noise case
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $f \in L^\infty(0,+ \infty;H)$ and let $R_h$ satisfy assumption iden-data-approx. Let $u$ be the solution to the equation deterministic-NS with initial condition $u_0 \in H$ and $U$ the solution to the equation deterministic-NS_nud with initial condition $U_0\in H$. Then, if $\mu$ and $h$ satisf where $c$ is a suitable positive constant, $c_0$ is the constant appearing in iden-data-approx and

Theorems & Definitions (41)

  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Lemma 4.3
  • proof
  • Corollary 4.4
  • proof
  • Proposition 4.5
  • proof
  • ...and 31 more