Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Martingale Suitable Weak Solutions of $3$-D Stochastic Navier-Stokes Equations with Vorticity Bounds

Weiquan Chen, Zhao Dong

TL;DR

This work develops a probabilistic framework for the 3D incompressible Navier–Stokes equations with non-linear noise by constructing martingale suitable weak solutions that satisfy stochastic local energy inequalities. The authors employ Leray regularization, prove tightness and use Skorokhod representation to pass to the limit, yielding a solution whose energy behavior is governed by an a.e. supermartingale and a Duchon–Robert–type local energy equality in the stochastic setting. They establish two stochastic local energy inequalities (LEI-I and LEI-II), identify the martingale structure controlling energy fluctuations, and provide a priori vorticity bounds when the initial vorticity is a finite measure. The results extend the deterministic dissipative-weak and Duchon–Robert frameworks to stochastic Navier–Stokes with general noise, without relying on OU-transformations, and offer rigorous local energy balance and regularity up to vorticity bounds.

Abstract

In this paper, we construct martingale suitable weak solutions for $3$-dimensional incompressible stochastic Navier-Stokes equations with generally non-linear noise. In deterministic setting, as widely known, ``suitable weak solutions'' are Leray-Hopf weak solutions enjoying two different types of local energy inequalities (LEIs). In stochastic setting, we apply the idea of ``martingale solution", avoid transforming to random system, and show new stochastic versions of the two local energy inequalities. In particular, in additive and linear multiplicative noise case, OU-processes and the exponential formulas DO NOT play a role in our formulation of LEIs. This is different to \cite{FR02,Rom10} where the additive noise case is dealt. Also, we successfully apply the concept of ``a.e. super-martingale'' to describe this local energy behavior. To relate the well-known ``dissipative weak solutions" come up with in \cite{DR00}, we derive a local energy equality and extend the concept onto stochastic setting naturally. For further regularity of solutions, we are able to bound the $L^\infty\big([0,T];L^1(Ω\times\mathbb T^3)\big)$ norm of the vorticity and $L^{\frac{4}{3+δ}}\big(Ω\times[0,T]\times\mathbb T^3\big)$ norm of the gradient of the vorticity, in case that the initial vorticity is a finite regular signed measure.

Martingale Suitable Weak Solutions of $3$-D Stochastic Navier-Stokes Equations with Vorticity Bounds

TL;DR

This work develops a probabilistic framework for the 3D incompressible Navier–Stokes equations with non-linear noise by constructing martingale suitable weak solutions that satisfy stochastic local energy inequalities. The authors employ Leray regularization, prove tightness and use Skorokhod representation to pass to the limit, yielding a solution whose energy behavior is governed by an a.e. supermartingale and a Duchon–Robert–type local energy equality in the stochastic setting. They establish two stochastic local energy inequalities (LEI-I and LEI-II), identify the martingale structure controlling energy fluctuations, and provide a priori vorticity bounds when the initial vorticity is a finite measure. The results extend the deterministic dissipative-weak and Duchon–Robert frameworks to stochastic Navier–Stokes with general noise, without relying on OU-transformations, and offer rigorous local energy balance and regularity up to vorticity bounds.

Abstract

In this paper, we construct martingale suitable weak solutions for -dimensional incompressible stochastic Navier-Stokes equations with generally non-linear noise. In deterministic setting, as widely known, ``suitable weak solutions'' are Leray-Hopf weak solutions enjoying two different types of local energy inequalities (LEIs). In stochastic setting, we apply the idea of ``martingale solution", avoid transforming to random system, and show new stochastic versions of the two local energy inequalities. In particular, in additive and linear multiplicative noise case, OU-processes and the exponential formulas DO NOT play a role in our formulation of LEIs. This is different to \cite{FR02,Rom10} where the additive noise case is dealt. Also, we successfully apply the concept of ``a.e. super-martingale'' to describe this local energy behavior. To relate the well-known ``dissipative weak solutions" come up with in \cite{DR00}, we derive a local energy equality and extend the concept onto stochastic setting naturally. For further regularity of solutions, we are able to bound the norm of the vorticity and norm of the gradient of the vorticity, in case that the initial vorticity is a finite regular signed measure.
Paper Structure (16 sections, 13 theorems, 222 equations)

This paper contains 16 sections, 13 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations
  4. Organization of the Paper
  5. Main Idea and Proof of Theorem \ref{['Thm. existence of martingale suitable solution']}
  6. Main ideas and propositions
  7. Existence of martingale suitable weak solutions
  8. Proof of Theorem \ref{['Thm. LEI. II']} and Theorem \ref{['Thm. energy supermartingale']}
  9. Local energy inequality (\ref{['LEI. II']})
  10. Super-martingale statement of local energy inequality
  11. Derivation of Local energy equality
  12. Proof of Proposition \ref{['Prop. Leray regularization']}, \ref{['Prop. LEE for Leray regularization']} and \ref{['Prop. vorticity bounds for Leray regularised solution']}
  13. Regularity estimates for Leray regularization
  14. Local energy balance for Leray regularized Navier-Stokes
  15. Vorticity bounds for Leray regularization
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{\varsigma_k\}$ satisfy (con. linear growth-con. uniform decay of the tails). Then for any given $r\in[1,\infty),~T>0$ and initial data $u_0\in L^2_\sigma$, there exists a martingale suitable weak solution $\{\left(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t\in[0,T]}, \mathbb{P}\right),u\}$ to ( with constant $C>0$.

Theorems & Definitions (30)

  • Remark 1
  • Remark 2
  • Definition 1
  • Definition 2: Martingale Suitable Weak Solutions
  • Remark 3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3: c.f. FR08
  • Theorem 1.4
  • ...and 20 more