Some Recent Developments

In this paper we derive a probabilistic representation of the deterministic 3-dimensional Navier--Stokes equations in the presence of spatial boundaries. The formulation in the absence of spatial boundaries was done by the authors in [Comm. Pure Appl. Math. 61 (2008) 330--345]. While the formulation in the presence of boundaries is similar in spirit, the proof is somewhat different. One aspect highlighted by the formulation in the presence of boundaries is the nonlocal, implicit influence of the boundary vorticity on the interior fluid velocity.

A stochastic-Lagrangian approach to the Navier--Stokes equations in domains with boundary

This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman [Probab. Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al. [Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140, 2004, in press]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.

Probability & incompressible Navier-Stokes equations: An overview of some recent developments

This is the publisher’s final pdf. The published article is copyrighted by the American Mathematical Society and can be found at: http://www.ams.org/publications/journals/journalsframework/proc.

/pdf/exponent-bounds-for-a-convolution-inequality-in-euclidean-34espamclb.pdf

Exponent bounds for a convolution inequality in Euclidean space with applications to the Navier-Stokes equations

The basic equations governing the motion of a fluid are well understood. For simplicity, we shall refer to the case of an incompressible, constant density, viscous Newtonian fluid; the velocity vector field $u\left (t,x\right )$ and pressure scalar field $p\left (t,x\right )$ satisfy the classical Navier–Stokes equations (in dimension 3) with viscosity ν > 0 

$$\displaystyle{ \begin{array}{rl} \partial _{t}u + (u \cdot \nabla )u + \nabla p& =\nu \varDelta u, \\ \mathop{\mathrm{div}}\nolimits u& = 0,\end{array} }$$

(15.1)

with appropriate initial and boundary conditions depending on the problem. For relatively simple fluid motions, these equations give us a very good tool for simulations and physical understanding. But there are complex fluid motions, those usually called turbulent, where special features are experimentally or numerically observed which do not have a clear explanation yet from the Navier–Stokes equations. In a sense, there is something at the foundation of fluid dynamics that is still unclear. For later reference, let us mention that this happens when a certain parameter R, called Reynolds number, is very large.

Cascade Representations for the Navier–Stokes Equations

A probabilistic approach int roduced by LeJan and Sznitrnan (1997) perm its derivation of weak solutions to 3-d incompressible Navier-Stokes equations whose Fourier transform may be rep resented by an expected value of a stochastic cascade. This approach was extended in Bhattacharya et al. (2003) by met hods which would yield unique global solutions by a stochastic representation under “small initial data conditions”. A connection to iterative contraction maps on appropriate function space was also provided which would also yield local existence and uniqueness under “short time” constraint s, but with out stochastic cascade representations. In t he present paper the authors (i.) Provide a stochastic cascade representation for local solutions, and (ii.) Provide time-asymptotics for global solutions from t he stochastic representation.

Semi-Markov Cascade Representations of Local Solutions to 3-D Incompressible Navier-Stokes

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Semi-Markov Cascade Representations of Local Solutions to 3-D Incompressible Navier-Stokes