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Full moment error estimates in strong norms for numerical approximations of stochastic Navier-Stokes equations with multiplicative noise, Part I: time discretization

Xiaobing Feng, Liet Vo

TL;DR

This work tackles the challenge of obtaining polynomial-order full moment error estimates in strong norms for the stochastic Navier-Stokes equations with multiplicative noise under Euler–Maruyama time discretization. It introduces a new analytical framework consisting of exponential stability, a discrete stochastic Gronwall inequality, a bootstrap argument, and a stochastic inf-sup framework to control nonlinear SPDEs with multiplicative noise. The authors prove optimal-order full moment error estimates for both velocity and pressure in energy norms, along with arbitrarily high-moment and pathwise results in the L^2-norm for the velocity, and establish a pressure error bound in a time-averaged L^2-norm. These results provide robust, quantitative error control for stochastic PDE numerics and lay groundwork applicable to broader nonlinear SPDEs.

Abstract

This paper focuses on deriving optimal-order full moment error estimates in strong norms for both velocity and pressure approximations in the Euler-Maruyama time discretization of the stochastic Navier-Stokes equations with multiplicative noise. Additionally, it introduces a novel approach and framework for the numerical analysis of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noise in general. The main ideas of this approach include establishing exponential stability estimates for the SPDE solution, leveraging a discrete stochastic Gronwall inequality, and employing a bootstrap argument.

Full moment error estimates in strong norms for numerical approximations of stochastic Navier-Stokes equations with multiplicative noise, Part I: time discretization

TL;DR

This work tackles the challenge of obtaining polynomial-order full moment error estimates in strong norms for the stochastic Navier-Stokes equations with multiplicative noise under Euler–Maruyama time discretization. It introduces a new analytical framework consisting of exponential stability, a discrete stochastic Gronwall inequality, a bootstrap argument, and a stochastic inf-sup framework to control nonlinear SPDEs with multiplicative noise. The authors prove optimal-order full moment error estimates for both velocity and pressure in energy norms, along with arbitrarily high-moment and pathwise results in the L^2-norm for the velocity, and establish a pressure error bound in a time-averaged L^2-norm. These results provide robust, quantitative error control for stochastic PDE numerics and lay groundwork applicable to broader nonlinear SPDEs.

Abstract

This paper focuses on deriving optimal-order full moment error estimates in strong norms for both velocity and pressure approximations in the Euler-Maruyama time discretization of the stochastic Navier-Stokes equations with multiplicative noise. Additionally, it introduces a novel approach and framework for the numerical analysis of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noise in general. The main ideas of this approach include establishing exponential stability estimates for the SPDE solution, leveraging a discrete stochastic Gronwall inequality, and employing a bootstrap argument.

Paper Structure

This paper contains 15 sections, 13 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and useful facts
  3. Space notation
  4. Some useful facts and inequalities
  5. Stochastic Navier-Stokes equations, weak solutions, and their properties
  6. Structure assumptions
  7. Weak formulation and solutions
  8. Properties of weak solutions
  9. Time approximation scheme
  10. Definition of the scheme
  11. Stability properties of the scheme
  12. Full moment error estimates in strong norms
  13. Basic (sub-second-order) full moment error estimates for the velocity approximation
  14. High moment and pathwise error estimates for the velocity approximation
  15. Second moment error estimates in the energy norm for the velocity and pressure approximations

Key Result

Lemma 2.1

Let $\{W(t):t\geq 0\}$ be a $\mathbb{R}$-valued Wiener process, then there holds where $C_m$ is a positive constant and $C_1 =1$.

Theorems & Definitions (22)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.1
  • Remark 3.1
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 12 more