# Titles and Abstracts

- Enrique Fernández Cara - Some controllability results in fluid mechanics

- John Gibbon - Quaternions and particle dynamics in the Euler fluid equations

rotations. It will be shown that they provide a natural way of selecting an appropriate ortho-normalframe -- designated the quaternion-frame -- for a particle in a Lagrangian flow, and of obtaining the

equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid

equations will then be considered.

- Martin Hairer - Long-time behaviour of the 2D stochastic Navier-Stokes equations

- Darryl Holm - Non-linear non-local equations for aggregation of particles carrying geometrical properties

- Dragos Iftimie - On the vanishing viscosity limit for Navier boundary conditions

- Alexander Kiselev - Global well-posedness for the critical 2D dissipative surface quasi-geostrophic equation

- Igor Kukavica - Conditional regularity and thin domain results for solutions of the Navier-Stokes equations

- Grzegorz Lukaszewicz - Turbulent shear flows and their attractors

In our research, motivated by applications in lubrication problems, we study twodimensional Navier-Stokes flows in channel-like domains and with various boundary conditions. We consider flows in both bounded and unbounded domains, and with both time independent and quite general time dependent forcing. Our aim is to prove existence of suitable attractors for a number of flows appearing in applications and to obtain estimates of dimension of the attractors in terms of parameters of the considered flows.

In particular, we are interested in influence of the geometry of the domain (physically, roughness of the surface) and boundary conditions (physically, character of boundary driving) of the flow on the attractor dimension.

We present also some recent abstract results on existence of attractors which prove useful in our research [2], [3], [5], [6] and some results about dimension of attractors, [1], [4]. We use, e.g., a version of the Lieb-Thirring inequality, in which constantsdepend explicitly on some norms representing geometry of the boundary [1]. References: [1] M. Boukrouche, G. Lukaszewicz, An upper bound on the attractor dimension of a 2D turbulent shear flow with a free boundary condition, Regularity and other aspects of the Navier-Stokes equations, Banach Center Publications Vol. 70, Warsaw 2005.

[2] M. Boukrouche, G. Lukaszewicz, On the existence of pullback attractor for a two-dimensional shear flow with Tresca’s boundary condition, submitted.

[3] T. Caraballo, G. Lukaszewicz & J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Analysis, TMA, vol.64, no.2, (2006), 484–498.

[4] J. Langa, G. Lukaszewicz & J. Real, Finite fractal dimension of pullback attractor for non-autonomous 2-D Navier-Stokes equations in some unbounded domains, Nonlinear Analisis, TMA, vol.66, (2007), 735–749.

[5] G. Lukaszewicz, Pullback attractors and statistical solutions for 2-D Navier- Stokes equations, to appear in DCDS-A.

[6] G. Lukaszewicz, A. Tarasi´nska, Pullback attractors for reaction-diffusion equation with unbounded right-hand side, in preparation.

- Josef Malek - On incompressible Navier-Stokes-Fourier equations and some of its generalizations

- Jose Rodrigo - Construction of Almost Sharp Fronts for the Surface Quasi-Geostrophic Equation

- Ricardo Rosa - Theory and applications of the statistical solutions of the Navier-Stokes equations

- Witold Sadowski - Numerical verification of regularity in the Navier-Stokes equations

Consider the 3D incompressible Navier-Stokes equations with zero forcing and periodic boundary conditions. It is known that for small enough initial data these equations have a regular solution. More precisely, for a fixed domain size and a given viscosity there exists constant C>0 such that all initial conditions with enstrophy less than C give rise to regular solutions. The value of the constant which follows from the theory of the Navier-Stokes equations is very small and in fact the enstrophy of all such solutions (those arising from initial conditions with enstrophy less than C) is decreasing in time. In the talk (based on the joint paper with James Robinson) I will present a numerical method which will verify, in a finite time, whether such a regularity result can be extended to all initial conditions with some arbitrary (but fixed) value of C.

- Maria Schonbek - Decay of Polymer equations and Poincaré estimates

Some questions in relation to Poincar\'e type inequalities, and fluid equations in general, will be discussed

- Marco Sammartino - Slightly viscous fluids: Well posedness results and blow-up phenomena

^{1}: in fact the presence of two counter-rotating vortices inside the boundary layer seem to produce a blow-up of the solution in an arbitrary short time.

We shall also discuss the situation when the initial datum for the 2D periodic Navier--Stokes equations are of the vortex layer type, in the sense that there is a rapid variation in the tangential component of the velocity across a curve. The vorticity is therefore concentrated in a layer whose thickness is of the order the square root of the viscosity. In the zero viscosity limit we derive (formally) the equations that rule the fluid inside the layer. Assuming the initial as well the matching (with the outer flow) data to be analytic, we shall prove that the model equations are well posed.