TY - JOUR
T1 - Stochastic Lagrangian formulations for damped Navier-Stokes equations and Boussinesq system, with applications
AU - Yamazaki, Kazuo
N1 - Publisher Copyright:
© Serials Publications Pvt. Ltd.
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
SN - 0973-9599
VL - 12
SP - 447
EP - 471
JO - Communications on Stochastic Analysis
JF - Communications on Stochastic Analysis
IS - 4
ER -