Global Existence and Stability of 3D Stochastic NSEs in Bounded and Unbounded Domains Driven by a Special Multiplicative Wiener Process
Abstract
The aim of this work is to extend the results from a recent paper by Hong, Li and Liu, from bounded domains to both bounded and unbounded domains. We show the global existence and uniqueness of 3D stochastic Navier-Stokes equations with nonlinear multiplicative noise for every initial data from the Sobolev space H1. We do not use any tightness argument. Instead, we firstly show the existence of a local maximal solution and then we use Lyapunov function to prove the solution is global. Our approach is motivated by a recent paper of the first named author with Ferrario, Maurelli and Zanella about a similar result for stochastic nonlinear Schrödinger Equations. The main idea is that once the local existence of strong solutions is established, a very strong noise pushing toward the origin, will make blow-up impossible.
Related
- Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative α-stable processes: Existence, uniqueness and numerical approximation
- Lyapunov-Krasovsky Functionals and the Existence of Unbounded Solutions of Functional-Differential Equations of Retarded Type
- On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative
- Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions
- Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data
BibTeX
@misc{brzezniak2026global,
title = {{Global Existence and Stability of 3D Stochastic NSEs in Bounded and Unbounded Domains Driven by a Special Multiplicative Wiener Process}},
author = {Zdzisław Brzeźniak and Shijia Zhang and Guoli Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08284},
eprint = {2609.08284},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08284},
}
RIS
TY - UNPB
TI - Global Existence and Stability of 3D Stochastic NSEs in Bounded and Unbounded Domains Driven by a Special Multiplicative Wiener Process
AU - Zdzisław Brzeźniak
AU - Shijia Zhang
AU - Guoli Zhou
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08284
UR - https://arxiv.org/abs/2609.08284
AB - The aim of this work is to extend the results from a recent paper by Hong, Li and Liu, from bounded domains to both bounded and unbounded domains. We show the global existence and uniqueness of 3D stochastic Navier-Stokes equations with nonlinear multiplicative noise for every initial data from the Sobolev space H^1. We do not use any tightness argument. Instead, we firstly show the existence of a local maximal solution and then we use Lyapunov function to prove the solution is global. Our approach is motivated by a recent paper of the first named author with Ferrario, Maurelli and Zanella about a similar result for stochastic nonlinear Schrödinger Equations. The main idea is that once the local existence of strong solutions is established, a very strong noise pushing toward the origin, will make blow-up impossible.
ER -