# Titles and Abstracts

• Claude Bardos - Besov Spaces and Euler Equation

In this talk I will try to show how Besov spaces are relevant to the the analysis of the time dependent Euler equation... In a series of contributions, E, Constantin and Titi, Pak and Park, Cheskidov, Friedlander, Shvydkoy and others they seem to provide sharp result for energy conservation and well posedeness. With the simple example of the shear flow I will show that the space $C^1$ is really critical. I intend also to show that the use of the space $\cal{B}^0_{\infty,1}$ seems compulsory to obtain global in time regularity of the 2d Euler equation with a non trivial 0 order perturbation. This is a joint work in progress with Anna Mazzucato and Edriss Titi motivated by some asymptotic regimes in 2d MHD.

• Michele Bartuccelli - Explicit (and sharp, hopefully) estimates for the dissipative length scale and corresponding global attractor dimension for the Swift-Hohenberg equation on the torus in one and two space-dimensions.

The Swift-Hohenberg equation is one of the most studied models in connection with patterns formation and localized structures. Proposed in 1977 by Swift and Hohenberg it has a major role in connection with Rayleigh–Bénard convection, Taylor–Couette flow, in the study of lasers and many other physics contexts. We view the Swift–Hohenberg equation as a model equation for a large class of higher-order parabolic model equations arising in a wide range of applications, such as the extended Fisher–Kolmogorov equation, the Cahn-Hilliard equation and equations with fourth order derivative terms arising in population dynamics. In this talk we will explicitly estimate its dissipative length scale and its attractor dimension in one and two spatial dimension on the torus.

• Luigi Berselli - On the vanishing viscosity limit for the 3D Navier-Stokes in bounded domains

We consider the vanishing viscosity limit for the Navier-Stokes under slip-without-friction boundary conditions and we study the problem in a general three-dimensional domain with non-flat boundary. We prove convergence in various strong norms, by means of a quite precise perturbation argument and the study of the vorticity equation.

• Alexey Cheskidov - On solutions of the 3D Navier-Stokes equations in the largest critical space

Recently there has been some interest in studying solutions of the 3D Navier-Stokes equations in the largest critical space. We will review some of the known results as well as present a new regularity criterion, which is weaker than every Prodi-Serrin condition, for instance.

• Sergei Chernyshenko - Global Stability Analysis of Fluid Flows Using Sum-of-Squares Polynomials

A new method is developed for assessing the stability of fluid flows within finite-dimensional approximations to the Navier-Stokes equations. Lyapunov functions, which are different from energy, are constructed using sum-of-squares optimization that exploits the property of energy conservation by the nonlinear terms in the Navier-Stokes equations. The technique is tested on a finite-dimensional model system. A method of reducing the dimension of the finite-dimensional approximation in this approach is proposed. This is a joint work with Paul J. Goulart.

• Peter Constantin - Remarks on Complex Fluids Models

I will discuss a few simplified models of particle-fluid systems based on the nonlinear Fokker-Planck equations coupled with fluid models. After briefly reviewing some of the known facts and basic open problems, I will describe a couple of global existence results, a blow up result and a baby model, all for the case in which configuration space of the particles is $R^d$.

• Diego Córdoba - Well-posedness for the Muskat problem

The Muskat problem involves filtration of two incompressible fluids throughout a porous medium. Recent work with collaborators has been focused on understanding the different features regarding well-posedness and regularity issues of the incompressible porous media equation. The aim of the talk is to describe some recent results on this problem.

• Charles Doering - Progress and Problems in the Analysis of (Turbulent) Energy Dissipation

In certain situations rigorous bounds on the time-averaged bulk energy dissipation rate in (weak) solutions of the 3D Navier-Stokes equations for incompressible Newtonian fluids are in qualitative (and sometimes nearly quantitative) accord with turbulence theory, simulations, and experiments. Nevertheless frustrating problems remain. For some physical set-ups including turbulent shear (Couette) flow between smooth plates, quantitative discrepancies such as (apparent) logarithmic corrections are unaccounted for in the current analysis. And on the other hand there are some simple boundary conditions and configurations for which no finite long-time averaged bounds are known at all. In this talk I will present some of these problems and propose two (endowed prize!) problems for the workshop participants to ponder.

• Charles Fefferman - Almost-sharp-front solutions of the suface QG equation

An "almost sharp front" is a temperature distribution with a large spatial gradient in a thin neighborhood of a smooth curve. We discuss how such an almost sharp front evolves in time by the surface quasigeostrophic equation. This crudely models the evolution of fronts in weather systems, and provides a simplified model of the evolution of vortex tubes by the 3D Euler equation. Joint work with Jose Rodrigo.

• Susan Friedlander - Advection-Diffusion Equations and Magnetogeostrophic Turbulence

We discuss an advection-diffusion equation that has been proposed by Keith Moffatt as a model for magnetogeostrophic turbulence in the Earth's fluid core. This nonlinear PDE (MG) has certain similarities to the critical surface quasi-geostrophic equation (QG), however it also has some crucial differences.
Inspired by the recent work of Caffarelli and Vasseur for the QG equation, we use De Giorgi techniques to prove Holder continuity for a class of active scalar equations where the divergence free velocity is in BMO-1. This general result implies that solutions of the MG equation, with L2 initial data, are smooth globally in time.
This work is joint with Vlad Vicol.

• Andrei Fursikov - Unboudedness of Stable Invariant Manifolds and Related Objects for Navier-Stokes System and Some Other Evolution PDE

In this talk an unbounded ellipsoid $El$ in Sobolev space $H^1$ will be constructed such that for each initial condition $v_0\in El$ there exists unique solution of 3D Navier-Stokes equations that exponentially decays at time $t\to\infty$ uniformly with respect to $v_0\in El$. Using this result the unboudedness of stable invariant manifold for 3D Navier-Stokes equations will be established. For certain model equations more precise description of unbounded domain in phase space that consists of exponentially decaying solution will be given.

• Isabelle Gallagher - Semiclassical and Spectral Analysis of Oceanic Waves

The aim of this talk is to describe the propagation of oceanic waves in the vicinity of the equator. More precesily we shall consider shallow water equations, subject to strong wind forcing and linearized around stationary profile, and we shall study the propagation of Rossby and Poincaré waves: the former develop closed trajectories whereas the latter are shown to disperse.This is a joint work with C. Cheverry, T. Paul and L. Saint-Raymond.

• Thierry Gallay - The stabilizing effect of fast rotation on two-dimensional vortices

Numerical simulations of freely decaying turbulence reveal that vortex interactions play a prominent role in the dynamics of two-dimensional viscous flows. Although basic phenomena such as vortex mergers are still beyond the scope of rigorous analysis, significant progress has been made recently in the understanding of the stability properties of a single isolated vortex. In particular, it is now possible to obtain quantitative estimates which show that fast rotation has a stabilizing effect on the vortex. This observation turns out to be crucial for the study of vortex interactions at high Reynolds numbers. The aim of this talk is to give an overview of the existing results in that direction, and to draw a few perspectives.

• John Gibbon - The Dynamics of a Gradient of Potential Vorticity Slides

In geophysical fluid dynamics (GFD) one of the most important quantities is potential vorticity (PV) defined by $q = \mbox{\boldmath\omega} \cdot\nabla\theta$ where $\mbox{\boldmath\omega}$ is the vorticity and $\theta$ the potential temperature. In non-dissipative flows q is conserved. However, little is understood about the dynamics of $\nabla q$ and $\nabla\theta$ which is essential for understanding of how PV accumulates in the oceans and atmosphere. In joint work with Darryl Holm this talk reports on an investigation of the transport of $\nabla{q}$ along surfaces of constant potential temperature $\theta$ in the stratified Euler, Navier-Stokes and the hydrostatic primitive equations of the oceans and atmosphere in terms of divergenceless flux vector $\mbox{\boldmath\mathcal{B}} = \nabla Q(q)\times\nabla\theta$, for any smooth function Q of q. The flux vector $\mbox{\boldmath\mathcal{B}}$ is shown to satisfy a transport equation

$\partial_{t}\mbox{\boldmath\mathcal{B}} - \mbox{curl}\,(\mbox{\boldmath\mathcal{U}} \times\mbox{\boldmath\mathcal{B}}) = - \nabla\big[qQ'(q)\,\mbox{div}\,\mbox{\boldmath\mathcal{U}}\big]\times\nabla\theta\,,$

where $\mbox{\boldmath\mathcal{U}}$ is a formal transport velocity. While the left hand side of this expression is reminiscent of that for magnetic filed flux in magnetohydrodynamics, the non-zero right hand side means that $\mbox{\boldmath\mathcal{B}}$ is not frozen into the flow when $\mbox{div} \,\mbox{\boldmath\mathcal{U}} \neq 0$. This result may apply to satellite observation of potential vorticity and potential temperature at the tropopause.

• Darryl Holm - Euler’s fluid equations: Optimal control vs optimization

The familiar Euler equations for incompressible flow of inviscid fluid are shown to follow from an optimisation problem that does not require that the Lagrangian particle labels exactly follow the flow lines of the Eulerian velocity vector field. Thus, a variational, or optimal control, problem and an optimisation problem for incompressible ideal fluid flow both yield the same Euler fluid equations, although their Lagrangian parcel dynamics are different. This is a result of the gauge freedom in the definition of the fluid pressure for an incompressible flow, in combination with the symmetry of fluid dynamics under relabelling of their Lagrangian coordinates. Similar ideas may be illustrated for SO(N) rigid body motion.

• Robert Kerr - Exploring necessary conditions for singularities using vortex dynamics

The long-term goal is to explain the TWISTS and KINKS in 3D DNS of the Euler and Gross-Pitaevskii equations using vortex dynamics. Both of these are ideal equations without viscous terms. This will be done by introduced a local vortex model that is a combination of anti-parallel Biot-Savart and the local induction approximation. A critical component is the torsion τ in Frenet-Serret formulation. The reason for going to the vortex model despite its limitations is that, while direct numerical simulations (DNS) are a useful tool for exploring the mathematics of continuum equations, by themselves, DNS cannot provide the answers This is especially true for the three-dimensional, incompressible Navier-Stokes and Euler equations. Questions to be asked include: What could bound singularities and what type of dynamics might be required for there to be a singularity of Euler?

• Igor Kukavica - Local well-posedness for a fluid-structure interaction model

In the talk we address a system of PDEs describing an interaction between an incompressible fluid and an elastic body. The fluid motion is modeled by the Navier-Stokes equations while an elastic body evolves according to an elasticity equation. On the common boundary, the velocities and stresses are matched. We discuss available results on local well-posedness and prove new existence and uniqueness results with the velocity and displacement belonging to low regularity spaces. The results are joint with A. Tuffaha and M. Ziane.

• Milton Lopes Filho - On the vortex-wave system

The vortex-wave system is the coupling of the two-dimensional vorticity equation with the point-vortex system. It is a mathematical model for the motion of regions of sharply concentrated vorticity in a general flow background, a situation of interest in geophysical flows, and in other applications. We review current knowledge regarding the vortex-wave system, with focus on an existence result for weak solutions with p-integrable background vorticity.

• Josef Málek - On implicitly constituted incompressible fluids Slides

We consider flows of incompressible fluids with a general implicit constitutive equation relating the deviatoric part of the Cauchy stress $\tens{S}$ and the symmetric part of the velocity gradient $\tens{D}$ in such a way that it leads to a maximal monotone (possibly multivalued) graph and the rate of dissipation is characterized by the sum of a Young function depending on $\tens{D}$ and its conjugate being a function of $\tens{S}$. Such a framework is very robust and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or Herschel-Bulkley type, and shear-rate dependent fluids with discontinuous viscosities as special cases. The appearance of the quantities $\tens{S}$ and $\tens{D}$ in all the assumptions characterizing the implicit relationship $\tens{G}(\tens{S}, \tens{D}) = \b0$ is fully symmetric. We are interested in large-data existence to steady/unsteady flows of such fluids completed by appropriate boundary and initial conditions, in both subcritical and supercritical cases. We use tools such Orlicz and Orlicz-Sobolev function spaces, properties of spatially dependent maximal monotone operators and Lipschitz approximations of Sobolev-Orlicz functions.

Based on a joint work with Miroslav Bulíček, Piotr Gwiazda and Agnieszka Świerczewska-Gwiazda.

• Helena Nussenzveig Lopes - On helical flows: vanishing viscosity limit and global existence for ideal fluids Slides

Helical flows are 3D flows which are covariant with respect to helical symmetry, in which simultaneous rotation and translation along the rotation axis occur. For incompressible helical flows it has been proved that there exist global strong solutions to the Navier-Stokes equations. Well-posedness for helical flows in the inviscid case has also been established under the assumption that the vorticity be bounded and that the helical swirl, the component of velocity along the helices, vanish. In this talk we will examine the vanishing viscosity limit of helical flows with finite enstrophy; we show, in several contexts, that the solutions converge to a solution of the inviscid problem, if the initial data has finite enstrophy and vanishingly small helical swirl. We also prove the existence of a weak helical flow, solution to the inviscid equations, with vorticity pth-power integrable, for some p>4/3, by using a different approximation. Finally, we comment on the two-dimensional limits of helical flows

• Gregory Seregin - Regularity Problem for the Navier-Stokes equations

In the talk recent regularity results for the energy solutions to the Navier-Stokes equations are going to be discussed.

• Roman Shvydkoy - Stationary singular solutions to the Euler equations

In this talk we explore a possibility of constructing solutions to the forced stationary Euler equations with limited regularity. The problem is motivated by finding vector fields that mimic the properties of a turbulent flow, i.e. anomalous energy dissipation, smooth forcing, and Holder continuity 1/3. The time-dependent version of this problem is known as the Onsager conjecture. We will exhibit a number of conditions which rule out existence of such solutions. Those include, for instance, conditions on the singularity set. On the other hand an example of an anomalous solution with smoothness 1/3, and integrability 18/11 will be presented.

• Alexis Vasseur - Regularity of solutions of some non linear integral variational problems

We will present the proof of the existence of classical solutions for a class of non-linear integral variational problems. Those types of equations are typically used in nonlocal image and signal processing. They involve nonlinear versions of fractional diffusion operators. The method is based on De Giorgi-Nash-Moser techniques. It extends to fully nonlinear settings a previous work on the regularity of solutions to the Surface Quasi-Geostrophic equations (SQG). This is a joint work with L. Caffarelli and Ch.-H. Chan.