TY - JOUR

T1 - Stochastic Lagrangian formulations for damped Navier-Stokes equations and Boussinesq system, with applications

AU - Yamazaki, Kazuo

PY - 2018

Y1 - 2018

N2 - We obtain stochastic Lagrangian formulations of solutions to some partial differential equations in fluid mechanics with diffusion, specifically damped Navier-Stokes equations, as well as the viscous and thermally diffusive Boussinesq system. As a byproduct of our discussion, we deduce stochastic Lagrangian formulations for other models, namely viscous and forced Burgers' equation, micropolar and magneto-micropolar fluid systems with zero vortex viscosity while positive and possibly distinct kinematic and angular viscosities, Bénard problem, as well as Leray-α magnetohydrodynamics model. Kelvin's circulation theorem is extended for the damped Navier- Stokes equations and the viscous and thermally diffusive Boussinesq system. The Cauchy formula for vorticity is extended from the damped Euler equations to the damped Navier-Stokes equations. The global well-posedness of the three-dimensional Euler equations with damping is proven for small initial data in critical Besov space. Finally, the global well-posedness of the four-dimensional Navier-Stokes equations with partial damping in only third and fourth components of the velocity field is also proven.

AB - We obtain stochastic Lagrangian formulations of solutions to some partial differential equations in fluid mechanics with diffusion, specifically damped Navier-Stokes equations, as well as the viscous and thermally diffusive Boussinesq system. As a byproduct of our discussion, we deduce stochastic Lagrangian formulations for other models, namely viscous and forced Burgers' equation, micropolar and magneto-micropolar fluid systems with zero vortex viscosity while positive and possibly distinct kinematic and angular viscosities, Bénard problem, as well as Leray-α magnetohydrodynamics model. Kelvin's circulation theorem is extended for the damped Navier- Stokes equations and the viscous and thermally diffusive Boussinesq system. The Cauchy formula for vorticity is extended from the damped Euler equations to the damped Navier-Stokes equations. The global well-posedness of the three-dimensional Euler equations with damping is proven for small initial data in critical Besov space. Finally, the global well-posedness of the four-dimensional Navier-Stokes equations with partial damping in only third and fourth components of the velocity field is also proven.

KW - Brownian motion

KW - Characteristics

KW - Feynman-Kac formula

KW - Kelvin's circulation theorem

KW - Lagrangian formulation

UR - http://www.scopus.com/inward/record.url?scp=85066732933&partnerID=8YFLogxK

U2 - 10.31390/cosa.12.4.05

DO - 10.31390/cosa.12.4.05

M3 - Article

AN - SCOPUS:85066732933

VL - 12

SP - 447

EP - 471

JO - Communications on Stochastic Analysis

JF - Communications on Stochastic Analysis

SN - 0973-9599

IS - 4

ER -