Global Existence and Non-Uniqueness for 3D Navier-Stokes Equations with Space-Time White Noise

We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier--Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most $-1/2-\kappa$ for any $\kappa>0$. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions.Our result applies to any divergence free initial condition in $L^{2}\cup B^{-1+\kappa}_{\infty,\infty}$, $\kappa>0$, and implies also non-uniqueness in law.

GLOBAL EXISTENCE AND NON-UNIQUENESS FOR 3D NAVIER–STOKES EQUATIONS WITH SPACE-TIME WHITE NOISE arXiv:2112.14093v1 [math.AP] 28 Dec 2021 MARTINA HOFMANOVÁ, RONGCHAN ZHU, AND XIANGCHAN ZHU Abstract. We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier–Stokes system driven by space-time white…

related reading

• A Feynman-Kac Approach for the Spatial Derivative of the Solution to the Wick Stochastic Heat Equation Driven by Time Homogeneous White Noisearxiv.org
• Stochastic Heat Equations with General Multiplicative Gaussian Noises: Hölder Continuity and Intermittencyarxiv.org
• Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditionsmath.emory.edu
• [2008.13092] Fundamental Solution to 1D Degenerate Diffusion Equation with Locally Bounded Coefficientsarxiv.org
• The Concentration of Invariant Measures for Stochastic Dynamical Systems with Locally Lipschitz Continuous Coefficients in R^darxiv.org
• Inverse Source Problems for the Stochastic Wave Equations: Far-Field Patternsarxiv.org
• A Primer on Stochastic Partial Differential Equationsmath.utah.edu
• Continuity in law with respect to the spatial Hurst index of the solutions to some linear SPDEs.pdfruor.uottawa.ca
• Density Dependent Singular Stochastic Differential Equationsarxiv.org
• Comparison principle for stochastic heat equation on Rdmath.emory.edu
• On Degenerate Partial Differential Equationsarxiv.org
• Asymptotic Stability of the Faraday Wave Problemtaisukeyasuda.github.io