Abstract. In this paper we derive a probabilistic representation of the deterministic 3-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 LANS-alpha models.

In this paper, we consider the modified quasi-geostrophic equationwith κ > 0, α ∈ (0, 1] and θ 0 ∈ L 2 (R 2 ). We remark that the extra Λ α−1 is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system.We assume κ > 0 and α ∈ (0, 1].Note that when α = 1 this is the critical dissipative quasi-geostrophic equation. The case of α = 0 arises when θ is the vorticity of a two dimensional damped inviscid incompressible fluid [3]. When κ > 0, α ∈ (0, 1), the dissipation term is the same as that of the supercritical quasi-geostrophic equation, however the extra Λ α−1 in the definition of u makes the drift term (u · ∇)θ scale the same way as the dissipation Λ α θ. Precisely, equations (1.3)-(1.4) are invariant with respect to the scaling θ ε (x, t) = θ(εx, ε α t), similar to the scaling invariance of the critical dissipative quasi-geostrophic equation.

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.When ν = 0, (1.1) and (1.2) are known as the Euler equations. These describe the evolution of the velocity field of an (ideal) inviscid and incompressible fluid. Formally the difference between the Euler and Navier-Stokes equations is only the dissipative Laplacian term. Since the Laplacian is exactly the generator a Brownian motion, one would expect to have an exact stochastic representation of (1.1) and (1.2) which is physically meaningful, that is, can be thought of as an appropriate average of the inviscid dynamics and Brownian motion.The difficulty, however, in obtaining such a representation is because of both the nonlinearity and the nonlocality of equations (1.1) and (1.2). In 2D, an exact stochastic representation of (1.1) and (1.2) dates back to Chorin [14] in 1973 and was obtained using vorticity transport and the Kolmogorov equations. In three dimensions, however, this method fails to provide an exact representation because of the vortex stretching term.In 3D, a variety of techniques has been used to provide exact stochastic representations of (1.1) and (1.2). One such technique (Le Jan and Sznitman [26]) uses a backward branching process in Fourier space. This approach has been extensively studied and generalized [3,4,32,35,36] by many authors (see also [37]). A different and more recent technique due to Busnello, Flandoli and Romito [6] (see also [5]) uses noisy flow paths and a Girsanov transformation. A related approach in [11] is the stochastic-Lagrangian formulation, exact stochastic representation of solutions to (1.1) and (1.2) which is essentially the averaging of noisy particle trajectories and the inviscid dynamics. Stochastic variational approaches (generalizing Arnold's [1] deterministic variational formulation for the Euler equations) have been used by [13,16] and a related approach using stochastic differential geometry can be found in [19].One common setback in all the above methods is the inability to deal with boundary conditions. The main contribution of this paper adapts the stochastic-Lagrangian formulation in [11] (where the authors only considered periodic boundary conditions or decay at infinity) to the situation with boundaries. The usual probabilistic techniques used to transition to domains with boundary involve stopping the processes at the boundary. This introduces two major problems with the techniques in [11]. First, stopping introduces spatial discontinuities making the proof used in [11] fail and a different approach is required....

Abstract. Consider a diffusion-free passive scalar θ being mixed by an incompressible flow u on the torus T d . Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field. Our main result shows that the mix-norm ( θ(t) H −1 ) is bounded below by an exponential function of time. The exponential decay rate we obtain is not universal and depends on the size of the support of the initial data. We also perform numerical simulations and confirm that the numerically observed decay rate scales similarly to the rigorous lower bound, at least for a significant initial period of time. The main idea behind our proof is to use recent work of Crippa and DeLellis ('08) making progress towards the resolution of Bressan's rearrangement cost conjecture.

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 L 2 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.