Abstract The aim of this contribution is to make a connection between two recent results concerning the dynamics of vortices in incompressible planar flows. The first one is an asymptotic expansion, in the vanishing viscosity limit, of the solution of the two dimensional Navier-Stokes equation with point vortices as initial data. In such a situation it is known, see Gallay (Arch. Ration. Mech. Anal.200 (2011) 445–490), that the solution behaves to leading order like a linear superposition of Oseen vortices whose centres evolve according to the point vortex system. However, higher order corrections can also be computed and these describe the deformation of the vortex cores due to mutual interactions. The second result is the construction by Smets & Van Schaftingen (Arch. Ration. Mech. Anal.198 (2010) 869–925) of “desingularized” solutions of the two-dimensional Euler equation. These solutions are stationary in a uniformly rotating or translating frame, and converge either to a single vortex or to a vortex pair as the size parameter ∈ goes to zero. We consider here the particular case of a pair of identical vortices, and we show that the solution of the weakly viscous Navier-Stokes equation is accurately described at time t by an approximate steady state of the rotating Euler equation which is a desingularized solution in the sense of Smets & Van Schaftingen (2010) with Gaussian profile and size

Introduction

Numerical simulations of freely decaying turbulence show that vortex interactions play a crucial role in the dynamics of two-dimensional viscous flows, see McWilliams (1984, 1990).

Abstract We review recent global existence and uniqueness results of solutions for models of complex fluids in ℝd. We describe results concerning the Oldroyd-B and related models.

Introduction

Complex fluids are ubiquitous in nature and manifest a rather large number of different behaviours. There is no single accepted general model for all these, and the presence of a large array of complicated models is an indication of the difficulties encountered at a fundamental level. In this article I will describe some of the mathematical issues. A complex fluid is a mixture between a solvent, which is treated as a normal fluid, and particulate matter in it. The particles are sufficiently many and sufficiently small compared to the characteristic scales of the motions of the solvent, so that one may hope for a description that does not have to resolve the fluid mechanical problem of flow past the particles. The particles themselves are treated in a simplified manner as objects m ∈ M, where M is a finite-degrees-of-freedom configuration space accounting for the salient features of the particles. For instance M can be a subset of ℝN or a more complicated metric space. Models have been devised to deal with microscopic elastic thread-like objects such as polymers (Doi & Edwards, 1998, Öttinger, 1996). The complicated hydrodynamic interactions are simplified using the separation of scales, replacing the many degrees of freedom due to them by a few representative ones.

Abstract The motion of the position of the maximum of vorticity ∥ζ∥∞ and growth of enstrophy are explored numerically using a recent calculation at moderately high resolution. We provide an exact analytic formula for the motion of ∥ζ∥∞ in the symmetry plane and use it to validate the numerical data. This motion drifts with respect to the local Lagrangian frame of reference due to the local gradient of vortex stretching, and can thus be associated with depletion of the circulation in the symmetry plane from the vicinity of ∥ζ∥∞. Despite this depletion, the numerical data is consistent with singular growth of the 3D enstrophy and symmetry-plane enstrophy that is bounded by inequalities analogous to the type known for the three-dimensional Navier-Stokes equations, using their respective enstrophies and with the viscosity being replaced by the circulation invariant.

Introduction

After the proof that the time integral of maximum vorticity controlled any possible singularities of the three-dimensional incompressible Euler equations (Beale, Kato, & Majda, 1984), it was realized that this was a quantity of low enough order to be accessible by numerical simulations, and the quest to determine numerically whether or not the Euler equations develop singularities ensued.

However, apparent singular growth in one quantity is insufficient for claiming consistency with singular growth as it is difficult to distinguish singular power-law growth from strong, but non-singular growth such as the exponential of an exponential. At a minimum, the growth in stretching near the position of maximum vorticity needs to be tracked independently and should grow in a manner consistent with the mathematical bounds, as first suggested by Pumir & Siggia (1990).

Abstract This paper considers the feedback stabilization problem for the Navier-Stokes equations defined in a bounded domain. Control via a forcing term (both distributed and impulsive) supported in a subdomain, via the initial condition, and via boundary conditions are studied, and relationships between these different kinds of control are presented. The precise meaning of ‘feedback control’ is discussed, and a feedback map providing control via the state variable is constructed for initial and distributed control (in the latter case for the linear Oseen equations only). Numerical algorithms for the calculation of stable invariant manifolds and projection operators on these sets are discussed. Finally the results of a numerical stabilization of a particular fluid flow are presented.

Introduction

The aim of this paper is to give a relatively short presentation of mathematical and numerical results concerning stabilization of the Navier-Stokes equations by feedback control. A description of the mathematical stabilization construction will be accompanied by a discussion of how mathematical notions of stabilization theory can be adapted to calculations.

Control theory for partial differential equations has been developed very intensively over the last few decades, and is now a very wide and farreaching field, even if one excludes extremal theory for partial differential equations. For some idea of this field more broadly, see the recent books by Coron (2007) and Tucsnak & Weiss (2009), as well as the earlier survey in Fursikov & Imanuilov (1999), together with the references in these publications.

The stabilization problem for the 2D Navier-Stokes system by a feedback distributed control supported in the whole fluid domain was studied by Barbu & Sritharan (1998).