Stochastic Lagrangian Path for Leray’s Solutions of 3D Navier–Stokes Equations

• Published in: Communications in mathematical physics 2021, Vol.381 (2), p.491-525
• Main Authors: Zhang, Xicheng, Zhao, Guohuan
• Format: Article
• Language: English
• Subjects: Brownian motion; Classical and Quantum Gravitation; Complex Systems; Fluid dynamics; Fluid flow; Markov processes; Mathematical analysis; Mathematical and Computational Physics; Mathematical Physics; Navier-Stokes equations; Particle trajectories; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Three dimensional motion; Velocity distribution
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03888-w

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray’s solution of 3D Navier–Stokes equations. More precisely, for any Leray’s solution u of 3D-NSE and each ( s , x ) ∈ R + × R 3 , we show the existence of weak solutions to the following SDE, which have densities ρ s , x ( t , y ) belonging to H q 1 , p with p , q ∈ [ 1 , 2 ) and 3 p + 2 q > 4 : d X s , t = u ( s , X s , t ) d t + 2 ν d W t , X s , s = x , t ⩾ s , where W is a three dimensional standard Brownian motion, ν > 0 is the viscosity constant. Moreover, we also show that for Lebesgue almost all ( s ,  x ), the solution X s , · n ( x ) of the above SDE associated with the mollifying velocity field u n weakly converges to X s , · ( x ) so that X is a Markov process in almost sure sense.