Stochastic Lagrangian Formulations for Damped Navier-Stokes Equations and Boussinesq System, with Applications

Abstract

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.

Global solutions to the damped MHD system

“…The well‐posedness of the MHD system has been studied by many mathematicians and great progress has been made in the past decades (see [3–5, 7–14, 17–21, 22–27]). Instead of the damping in (1.1), the MHD system involving fluid viscosity and magnetic diffusion is in the following form: $$$$\begin{eqnarray} {\left\lbrace \begin{aligned} &\partial_t u -\mu \Delta u+u \cdot \nabla u+\nabla {P}-b \cdot \nabla b=0,\\[-4pt] &\partial_t b -\nu \Delta b+u \cdot \nabla b-b \cdot \nabla u=0, \\[-4pt] & \hbox{\rm div}\, u= \hbox{\rm div}\, b=0,\\[-4pt] &(u,b)|_{t=0}=\big (u_0,b_0\big ), \end{aligned}\right.}$$…”

“…The system (1.1) is a well-known MHD model which governs the dynamics of the velocity and magnetic fields in an electrically conducting fluid such as plasma, liquid metal, salt water, etc. Especially, when we neglect the effect of the magnetic field, (1.1) reduces to be the damped Euler equations (see [2,15,21,23,28]).…”

The stochastic tamed MHD equations: existence, uniqueness and invariant measures

“…You [52] proved existence of a random attractor under the assumption of well-posedness for additive noise for β ∈ (3,5] (notably leaving out the critical, or tamed, case β = 3). Yamazaki [49] proved a Lagrangian formulation and extended Kelvin's circulation theorem to the partially damped case (i.e. only a few components are damped, but the damping there is much stronger, e.g.…”