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Almost global existence for the stochastic Navier-Stokes equations with small $H^{1/2}$ data

Mustafa Sencer Aydın, Igor Kukavica, Fanhui Xu

TL;DR

This work studies global existence for the stochastic Navier-Stokes equations with multiplicative noise on $\mathbb{T}^3$ for initial data in $H^{1/2}$, achieving almost global-in-time solutions with high probability when data and noise are small. A key innovation is an infinite decomposition of the initial data into smoother components $u_0=\sum_{k\ge0} v_0^{(k)}$, enabling a cascade of SNSE-type problems with cutoffs and a fixed-point scheme to control the nonlinear advection via $H^{\frac12+\delta}$ regularity. The authors prove global energy bounds and probabilistic stopping-time controls, then pass to the limit to obtain a probabilistically strong solution up to a stopping time $\tau$, with $\mathbb{P}(\tau=\infty)\ge 1-p_0$ under small data/noise, and a local-in-time existence result when the noise is not necessarily small. The approach leverages the stochastic heat equation as a core auxiliary tool to manage regularity and energy, offering a novel pathway around criticality for SNSE with multiplicative noise. These results advance understanding of stochastic fluid models at critical regularity and provide quantitative probabilistic guarantees for global behavior in low-regularity regimes.

Abstract

We address the global existence of solutions to the stochastic Navier-Stokes equations with multiplicative noise and with initial data in $H^{1/2}(\mathbb{T}^{3})$. We prove that the solution exists globally in time with probability arbitrarily close to~$1$ if the initial data and noise are sufficiently small. If the noise is not assumed to be small, then the solution is global on a sufficiently small deterministic time interval with probability arbitrarily close to~$1$.

Almost global existence for the stochastic Navier-Stokes equations with small $H^{1/2}$ data

TL;DR

This work studies global existence for the stochastic Navier-Stokes equations with multiplicative noise on for initial data in , achieving almost global-in-time solutions with high probability when data and noise are small. A key innovation is an infinite decomposition of the initial data into smoother components , enabling a cascade of SNSE-type problems with cutoffs and a fixed-point scheme to control the nonlinear advection via regularity. The authors prove global energy bounds and probabilistic stopping-time controls, then pass to the limit to obtain a probabilistically strong solution up to a stopping time , with under small data/noise, and a local-in-time existence result when the noise is not necessarily small. The approach leverages the stochastic heat equation as a core auxiliary tool to manage regularity and energy, offering a novel pathway around criticality for SNSE with multiplicative noise. These results advance understanding of stochastic fluid models at critical regularity and provide quantitative probabilistic guarantees for global behavior in low-regularity regimes.

Abstract

We address the global existence of solutions to the stochastic Navier-Stokes equations with multiplicative noise and with initial data in . We prove that the solution exists globally in time with probability arbitrarily close to~ if the initial data and noise are sufficiently small. If the noise is not assumed to be small, then the solution is global on a sufficiently small deterministic time interval with probability arbitrarily close to~.
Paper Structure (7 sections, 11 theorems, 127 equations)

This paper contains 7 sections, 11 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the main results
  3. Stochastic heat equation
  4. Global solutions of the truncated difference equations
  5. Decomposition into smooth data and a construction of $H^{\frac{1}{2} +\delta}$ solutions
  6. The global energy control and pointwise estimates
  7. Conclusion of the proofs of Theorems \ref{['T01']} and \ref{['T02']}

Key Result

Theorem 2.1

Let ${{u}}_0\in L^\infty(\Omega; H^{1/2}(\mathbb T^3))$ be such that $\nabla\cdot u_0=0$ and $\int_{\mathbb T^3} u_0=0$. Suppose that the assumptions EQ04 and EQ05 hold, where $\epsilon_{\sigma}\in(0,1]$ is sufficiently small. For every $p_0\in(0,1]$, there exists $\epsilon_0\in(0,1]$ such that if then there is a stopping time $\tau\in(0,\infty]$ and a unique solution $(u, \tau)$ of EQ01 on $(\Om

Theorems & Definitions (22)

  • Theorem 2.1: Global solution with small noise and initial data
  • Theorem 2.2: Local solution with small initial data
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['L01']}
  • Lemma 4.1: Decomposition of initial data
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['LM01']}
  • Lemma 4.3
  • proof : Proof of Lemma \ref{['LM01']}
  • Lemma 4.4: An $H^{\frac{1}{2}}$-energy control
  • ...and 12 more