Gautam Iyer

A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations

In this paper we derive a probabilistic representation of the deterministic three-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self-contained proof of local existence for the nonlinear stochastic system and can be extended to formulate stochastic representations of related hydrodynamic-type equations, including viscous Burgers equations and Lagrangian-averaged Navier-Stokes alpha models.

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

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.

Dynamic cubic instability in a 2D Q-tensor model for liquid crystals

We consider a four-elastic-constant Landau–de Gennes energy characterizing nematic liquid crystal configurations described using the Q-tensor formalism. The energy contains a cubic term and is unbounded from below. We study dynamical effects produced by the presence of this cubic term by considering an L2 gradient flow generated by this energy. We work in two dimensions and concentrate on understanding the relations between the physicality of the initial data and the global well-posedness of the system.

EXIT TIMES OF DIFFUSIONS WITH INCOMPRESSIBLE DRIFT

Let

A stochastic-Lagrangian particle system for the Navier–Stokes equations

\Omega\subset\mathbb R^n

A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

be a bounded domain and for

Anomalous diffusion in fast cellular flows at intermediate time scales

x\in\Omega

Coercivity and stability results for an extended Navier-Stokes system

let

Numerical studies of homogenization under a fast cellular flow

\tau(x)

A Model for Vortex Nucleation in the Ginzburg–Landau Equations

be the expected exit time from