Nonuniqueness in law for stochastic hypodissipative Navier–Stokes equations. (February 2023)

• Record Type: Journal Article
• Authors: Rehmeier, Marco; Schenke, Andre
• Is Part Of: Nonlinear analysis. Volume 227(2023)
• Publication Date: 2023-02
• Subjects: 60H15 -- 35R60 -- 35Q35 -- 35R25; Stochastic partial differential equations -- Fractional Navier–Stokes equations -- Convex integration -- Nonuniqueness -- Martingale solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113179
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the incompressible hypodissipative Navier–Stokes equations with dissipation exponent 0 < α < 1 2 on the three-dimensional torus perturbed by an additive Wiener noise term and prove the existence of an initial condition for which distinct probabilistic weak solutions exist. To this end, we employ convex integration methods to construct a pathwise probabilistically strong solution, which violates a pathwise energy inequality up to a suitable stopping time. This paper seems to be the first in which such solutions are constructed via Beltrami waves instead of intermittent jets or flows in a stochastic setting.