Markov selections for the 3D stochastic Navier–Stokes equations

• Published in: Probability theory and related fields Vol. 140; no. 3-4; pp. 407 - 458
• Main Authors: Flandoli, Franco, Romito, Marco
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-03-2008; Springer; Springer Nature B.V
• Subjects: Boundary conditions; Continuity; Economics; Energy; Exact sciences and technology; Finance; Fluid flow; General topics; Initial conditions; Insurance; Linear equations; Management; Markov analysis; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Noise; Operations Research/Decision Theory; Parametric inference; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Quantitative Finance; Sciences and techniques of general use; Statistics; Statistics for Business; Stochastic analysis; Studies; Theoretical; Three dimensional imaging; Markov selections; 76M35; Markov property; Strong Feller property; Well posedness; 60H30; Stochastic Navier–Stokes equations; 76D05; 35Q30; 60H15; Martingale problem; Additive noise; Initial condition; Well posedness, 76D05; Probability theory; Feller property; Sufficient condition; Navier Stokes equation; Stochastic Navier-Stokes equations; Stochastic equation
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-007-0069-y

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier–Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier–Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.