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Nonuniqueness in law of stochastic 3d navierstokes equations with general multiplicative noise

Huaxiang Lv, Yichun Zhu

TL;DR

This work establishes non-uniqueness in law for the three-dimensional stochastic Navier–Stokes equations on $\mathbb{T}^3$ with general multiplicative noise $G(u)\,dW$, by developing a stochastic convex integration framework. Under a structural assumption on $G$, for any divergence-free initial data in $L^2$ there exist infinitely many analytically weak and probabilistically strong global solutions, together with infinitely many ergodic stationary solutions, the latter obtained via Krylov–Bogoliubov arguments. The construction combines a stochastic splitting into a linear stochastic convolution and a nonlinear remainder, with a carefully designed random oscillatory perturbation built from intermittent jets to cancel Reynolds stresses while keeping control of energy. This demonstrates ill-posedness and rich long-time behavior under broad noise mechanisms, extending known results beyond additive or highly regular noise to a wide class of multiplicative noises with potential implications for turbulence modeling and stochastic PDE theory.

Abstract

We are concerned with the three dimensional navier-stokes equations driven by a general multiplicative noise. For every divergence free and mean free initial condition in L2, we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, which implies non-uniqueness in law. Moreover, we prove the existence of infinitely many ergodic stationary solutions. Our results are based on a stochastic version of the convex integration and the Ito calculus.

Nonuniqueness in law of stochastic 3d navierstokes equations with general multiplicative noise

TL;DR

This work establishes non-uniqueness in law for the three-dimensional stochastic Navier–Stokes equations on with general multiplicative noise , by developing a stochastic convex integration framework. Under a structural assumption on , for any divergence-free initial data in there exist infinitely many analytically weak and probabilistically strong global solutions, together with infinitely many ergodic stationary solutions, the latter obtained via Krylov–Bogoliubov arguments. The construction combines a stochastic splitting into a linear stochastic convolution and a nonlinear remainder, with a carefully designed random oscillatory perturbation built from intermittent jets to cancel Reynolds stresses while keeping control of energy. This demonstrates ill-posedness and rich long-time behavior under broad noise mechanisms, extending known results beyond additive or highly regular noise to a wide class of multiplicative noises with potential implications for turbulence modeling and stochastic PDE theory.

Abstract

We are concerned with the three dimensional navier-stokes equations driven by a general multiplicative noise. For every divergence free and mean free initial condition in L2, we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, which implies non-uniqueness in law. Moreover, we prove the existence of infinitely many ergodic stationary solutions. Our results are based on a stochastic version of the convex integration and the Ito calculus.
Paper Structure (36 sections, 25 theorems, 307 equations)

This paper contains 36 sections, 25 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. Linear multiplicative noise
  5. Matrix-valued linear multiplicative noise
  6. Multiplicative noise with Lipschitz continuous coefficients
  7. Preliminaries
  8. Notations
  9. The operator $\Pi_{\zeta}$
  10. Inverse divergence operators $\mathcal{R}$ and $\mathcal{B}$
  11. Stochastic convolution
  12. Construction of non-unique stationary solutions
  13. Notations on parameters and iteration set up
  14. Proof of Theorem \ref{['thm:main:1']}
  15. Proof of Theorem \ref{['thm:main:2']}
  16. ...and 21 more sections

Key Result

Theorem 1.2

Suppose Assumption a:G:1 holds. Let $u_0 \in L^2_{\sigma}$, $\mathbf{P}-a.s.$, be independent of the cylindrical Wiener process $W_t$, $t \geqslant 0$. There exist infinitely many analytically weak and probabilistically strong solutions to 1 in the sense of Definition d:sol. The solutions belong to

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 1.9
  • proof
  • ...and 46 more