# Titles and Abstracts

• Enrique Fernández Cara - Some controllability results in fluid mechanics
This talk is devoted to present some recent results concerning the controllability of some linear and nonlinear equations from fluid mechanics. We will analyze the local and global exact controllability to bounded trajectories. We will first deal with the Burgers equation. Some positive and negative results will be given in this case. Then, we will consider the Navier-Stokes and other related systems.

• John Gibbon - Quaternions and particle dynamics in the Euler fluid equations
More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis
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
One of the cleanaest mathematical models of two-dimensional turbulence are the stochastic Navier-Stokes equations. We give sufficient (and in some sense close to necessary) conditions on the covariance of the driving force to obtain the uniqueness of the stationary state of these equations. It can be shown that under these conditions, the convergence in law of arbitrary solutions to the stationary one is exponential. As a consequence, one shows that the generator of the dynamics has a spectral gap in a suitable space of observables.
• Darryl Holm - Non-linear non-local equations for aggregation of particles carrying geometrical properties
Aggregation of particles whose interaction potential depends on their mutual orientation is considered. The aggregation dynamics is derived using a version of Darcy's law and a variational principle depending on the geometric nature of the physical quantities. The evolution equation that results from this procedure is a combination of a nonlinear diffusion equation and a double bracket equation. The Landau-Lifshitz equation is obtained as a particular case. We also derive analytical solutions of equations which are collapsed (clumped) states and show their dynamical emergence from smooth initial conditions in numerical simuations. Finally, we compare a numerical solution of our equation with recent experiments on self-assembly of star-shaped pbjects floating on the surface of water (P.D.Weidman).

• Dragos Iftimie - On the vanishing viscosity limit for Navier boundary conditions
We discuss the issue of the inviscid limit of the incompressible Navier-Stokes equations when the Navier boundary conditions are prescribed. We justify an asymptotic expansion which involves a weak amplitude boundary layer, with the same thickness as in Prandtl's theory and a linear behavior. This analysis holds for general regular domains, in both dimensions two and three.

• Alexander Kiselev - Global well-posedness for the critical 2D dissipative surface quasi-geostrophic equation
Surface quasi-geostrophic equation arises in modelling rotating fluids and is relevant to studies of atmosphere and ocean. On the mathematical level, the equation can be thought of as a model intermediate between Burgers and 3D Navier-Stokes equations. Two cases are of special interest: conservative and with critical (square root of Laplacian) dissipative term. We give an elementary proof of the global well-posedness for the critical 2D dissipative surface quasi-geostrophic equation. The argument is based on a new non-local maximum principle involving appropriate moduli of continuity. The talk is based on a joint work with Fedja Nazarov and Alexander Volberg.

• Igor Kukavica - Conditional regularity and thin domain results for solutions of the Navier-Stokes equations
We consider sufficient conditions for regularity f Leray-Hopf solutions of the Navier-Stokes equation. By a result of Neustupa and Panel, a Leray-Hopf weak solution is regular provided a single component of the velocity is bounded. In this talk we will survey existing and present new results on one component and one direction regularity. We will also show global regularity for a class of solutions of the Navier-Stokes equation in thin domains. This is a joint work with M. Ziane.
• Grzegorz Lukaszewicz - Turbulent shear flows and their attractors
Existence of attractors and estimates of their dimension for shear turbulent flows have been studied in many papers. They follow earlier investigations on boundary driven flows between parallel plates, stability of Couette flow, and the onset of turbulence.
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
In the first part, we consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the the balance of mass, the balance of linear momentum and the balance of energy, and is completed by the entropy inequality. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier's slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions. It has been well documented that the viscosity and the thermal conductivity of most liquids depend also on the pressure, and the shear rate. The relevant experimental studies show that even at high pressures the variations of the values in the density are insignificant in comparison to that of the viscosity, and it is thus reasonable to assume that the liquids in question are incompressible fluids with pressure, shear rate and tmperature dependent viscosities. In the second part of the talk, we discuss physical issues relevant to such fluids and present the mathematical properties concerning unsteady three-dimensional internal flows of such incompressible fluids. Assuming that we have Navier's slip at the impermeable boundary we establish the long-time existence of a (suitable) weak solution when the data are large.
• Genevieve Raugel - A hyperbolic pertubation of the Navier-Stokes equations

• Jose Rodrigo - Construction of Almost Sharp Fronts for the Surface Quasi-Geostrophic Equation
Sharp fronts for the surface quasi-geostrophic equation are the analogue of vortex lines for 3D Euler. We present a construction of almost sharp-fronts (the analogue of vortex tubes for 3D Euler) of any (small) thickness, for which the time of existence is bonded below by a constant independent of the thickness. This result, together with previous work of Cordoba, Fefferman and Rodrigo provides a rigorous derivation of the equation for a sharp front that only involves tools avaialble in 3D Euler. This is joint work with Charles Fefferman.

• Ricardo Rosa - Theory and applications of the statistical solutions of the Navier-Stokes equations
The concept of statistical solution is akin to the notion of ensemble average in the statistical theory of turbulence and is relevant to the mathematical theory of turbulent flows. We present some recent applications of such statistical solutions in the derivation of rigorous bounds for physical quantities associated with channel flows driven by a uniform pressure gradient. We also discuss some new results and open problems for such solutions in a more abstract sense.

• 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
We study the decay and existence of solutions to some equations modeling polymeric flow. We consider the case when the drag term is corotational and the solutions are sufficiently regular to satisfy some necessary energy estimates. We analyse the decay when when the space of elongations is bounded, and the spatial domain of the polymer is either a bounded domain $\Omega \subset \mathbb{R}^n, n=2,3$ or the domain is the whole space $\mathbb{R}^n, n=2,3$. The decay is first established for the probability density $\psi$ and then this decay is used to obtain decay of the velocity $u$. Consideration also is given to solutions where the probability density is radial in the admissible elongation vectors $q$. In this case the velocity $u$, will become a solution to Navier-Stokes equation, and thus decay follows from known results for the Navier-Stokes equations.
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
When a high Reynolds number fluid interacts with a rigid boundary one can derive Prandtl's equations as a formal asymptotic limit of the Navier-Stokes equations. In this talk we shall review some known short time results for Prandtl's equations and investigate the process leading to the formation of a singularity in the solution. Moreover we shall show some numerical evidence of the ill posedness of Prandtl's equation in H1: 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.