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On conditional uniqueness of solutions to stochastic Navier-Stokes Equations

István Gyöngy, Nicolai V. Krylov

TL;DR

This work extends conditional uniqueness results for the deterministic Navier–Stokes equations to the stochastic setting in $\mathbb{R}^d$ with $d\ge3$. By introducing admissible random solutions that decompose as $u = u^M + u^B$ with a Morrey-component $u^M\in E_{r,1}$ and a bounded-in-space component $u^B\in L_2([0,T],L_\infty)$, the authors prove the existence of a time-continuous $\mathcal{H}$-valued modification (Theorem 1) and establish conditional uniqueness under a small Morrey-norm bound $\hat{u}<\delta/N$ (Theorem 2). A corollary recovers the Ladyzhenskaya–Prodi–Serrin uniqueness criterion in this stochastic admissible framework, and Theorem 3 provides conditional regularity results under additional smoothness assumptions. The results generalize Prodi–Serrin uniqueness to stochastic Navier–Stokes and offer a broader functional-analytic setting via Morrey-admissible functions, with potential implications for well-posedness and regularity in stochastic fluid dynamics.

Abstract

This paper is a continuation of [26]. Here theorems on conditional uniqueness and regularity for solutions to stochastic Navier-Stokes equations in $\mathbb R^d$ are presented.

On conditional uniqueness of solutions to stochastic Navier-Stokes Equations

TL;DR

This work extends conditional uniqueness results for the deterministic Navier–Stokes equations to the stochastic setting in with . By introducing admissible random solutions that decompose as with a Morrey-component and a bounded-in-space component , the authors prove the existence of a time-continuous -valued modification (Theorem 1) and establish conditional uniqueness under a small Morrey-norm bound (Theorem 2). A corollary recovers the Ladyzhenskaya–Prodi–Serrin uniqueness criterion in this stochastic admissible framework, and Theorem 3 provides conditional regularity results under additional smoothness assumptions. The results generalize Prodi–Serrin uniqueness to stochastic Navier–Stokes and offer a broader functional-analytic setting via Morrey-admissible functions, with potential implications for well-posedness and regularity in stochastic fluid dynamics.

Abstract

This paper is a continuation of [26]. Here theorems on conditional uniqueness and regularity for solutions to stochastic Navier-Stokes equations in are presented.

Paper Structure

This paper contains 5 sections, 11 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Formulation of the main theorems
  3. Preliminaries
  4. Proof of Theorems \ref{['theorem 1']} and \ref{['theorem 2']}
  5. On $W^1_2$-regularity of admissible solutions

Key Result

Theorem 2.1

Let Assumptions assumption 1, assumption 2 (i) and assumption 3 hold. Assume $u$ is an admissible solution to eq-ini. Then $u$ is an $\mathcal{H}$-solution in the sense that it has a $P\otimes dt$-modification, denoted also by $u$, which is an $\mathcal{H}$-solution. Moreover, where $N=N(d,\delta)$ is a constant and with a constant $N'=N'(\delta)$ and any predictable process $\mu\geq0$ such that

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • ...and 18 more