Absolute continuity of the law for the two dimensional stochastic\n Navier-Stokes equations

2017/02/06 by Benedetta Ferrario, Ferrario, Benedetta, Margherita Zanella +1
Economics, Econometrics and Finance · Engineering · Mathematics · #35Q30 #60H07 #60H15 #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1702.01597

openalex publication_date 2017/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the two dimensional Navier-Stokes equations in vorticity form\nwith a stochastic forcing term given by a gaussian noise, white in time and\ncoloured in space. First, we prove existence and uniqueness of a weak (in the\nWalsh sense) solution process \ξ and we show that, if the initial vorticity\n\ξ₀ is continuous in space, then there exists a space-time continuous\nversion of the solution. In addition we show that the solution \ξ(t,x)\n(evaluated at fixed points in time and space) is locally differentiable in the\nMalliavin calculus sense and that its image law is absolutely continuous with\nrespect to the Lebesgue measure on \ℝ.\n

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