# Titles and Abstracts

• Claude Bardos (Paris)

Title: On boundary effect for zero viscosity limit of solutions of Navier-Stokes
Abstract: This contribution is based on the Kato theorem and on some work which I started with Edriss Titi and continued with Wiedemann and Szekelyhidi.To the best of my knowledge the Kato theorem provides the simplest relation between the anomalous energy dissipation and the appearance of turbulence. Both if and only if conditions. For such anomalous energy dissipation both if and only if conditions and sufficient conditions have been provided.But what is striking is that these conditions are in full agreement with observations coming from engineering science or fluid mechanic litterature.

• Yann Brenier (Ecole Polytechnique)

Title: On the Vlasov-Monge-Ampere system
Abstract: About twelve years ago, the Vlasov-Monge-Ampere system was shown (by Loeper and myself) to be a suitable approximation of the Euler equations of incompressible fluids.
This idea has known a revival in the last few years, in particular by Merigot and collaborators, at the numerical level, and by myself through a new derivation of the Vlasov-Monge-Ampere system from the very elementary stochastic model of a brownian cloud of indistinguishable particles without interaction.

• Antoine Choffrut (Warwick)

Title: On the global structure of the set of 2D stationary Euler flows
Abstract: The incompressible Euler equations govern the evolution of an ideal fluid. They enjoy a very elegant geometric Hamiltonian formulation and as such their steady-states are of particular interest. Being an infinite-dimensional system, the analysis becomes considerably more challenging and the usual toolbox is not adequate for the Euler equations. In this talk I will present some recent work with Herbert Koch on the existence of steady-states with prescribed vorticity distribution. It is a global version of previous work which I did with Vladimir Sverak. One crucial ingredient is to derive sufficiently strong a priori estimates. I will also discuss other interesting aspects of the proof.

• Jean-Yves Chemin (Université Pierre et Marie Curie)

Title: Some large data global existence results for inhomogeneous incompressible viscous
Abstract: In this talk, we want to make some review about classical results about local and global wellposedness for the incompressible 2D ans 3D Navier-Stokes equations with variable density (i.e. inhomogeneous) with no vaccum, namely global wellposedness for 2D fluids and local wellposedness and global for small data in 3D fluids. We intend to explain how the concept of slowly varying fluid with respect to one variable is able to provide a large class of global solutions which are not related to small initial data.

• Diego Córdoba (ICMAT - Madrid)

Title: Stability shifting and mixing solutions for the Muskat problem
Abstract: The Muskat equation governs the motion of an interface separation of two incompressible fluids in a porous media. In this talk I will present the following recent results:
(1) Global existence for a new family of initial data (joint work with O. Lazar).
(2) The existence of solutions which shift stability regimes in the following sense: they start stable, then become unstable, and finally return back to the stable regime before it breaks down (joint work with J. Gomez-Serrano and A. Zlatos).
(3) The existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime (joint work with A. Castro and D. Faraco).

• David Gerard-Varet (Université Paris Diderot)

Title: Justification of boundary layer asymptotics in the 2D Navier-Stokes equation
Abstract: We will present a recent joint work with Maekawa and Masmoudi on the justification of boundary layer expansions for solutions of 2D Navier-Stokes. This justification depends much on the regularity and monotonicity properties of the initial data, due to subtle instability phenomena in the boundary layer. We will discuss these phenomena, and provide an optimal short time stability result in Gevrey class, improving a result of Sammartino and Caflisch in the analytic setting.

• John Gibbon (Imperial College)

Title: Analysis of PDEs involving the Rayleigh-Taylor instability
Abstract: The phenomenon of the Rayleigh-Taylor instability (RTI), involving the mixing of a heavy fluid overlaying a lighter, has a long history, not least because it occurs in laboratory tank and plasma fusion experiments, multi-phase physics and astro-physical phenomena. The approach taken by different schools varies widely. We will briefly review this work but spend most of the talk on what is known as the variable density model (VDM) pioneered by Cook and Dimotakis (2001) and Livescu and Ristorcelli (2007). The PDEs, coupled to the 3D Navier-Stokes equations, have a remarkably elegant nature but pose hard technical challenges. Our analysis uses the data generated by Daniel Livescu (LANL) and available on the Johns Hopkins Turbulence data-base (JHTDB). If time permits I will briefly discuss the occurrence of the RTI at interfaces in multi-phase physics.

• Robert Jerrard (Toronto)

Title: Vortex Filaments in the Euler Equation
Abstract: Classical fluid dynamics arguments suggest that in certain limits, the evolution of thin vortex filaments in an ideal incompressible fluid should roughly be governed by an equation called the binormal curvature flow. However, these classical arguments rely on assumptions that are so unrealistic that it would be hard even to extract from them a precise conjecture that admits any realistic possibility of a proof. We present a different approach to this question that yields a reasonable formulation of a conjecture and genuinely plausible supporting evidence, and that clarifies the very substantial obstacles to a full proof. Parts of the talk are based on joint work with Didier Smets and with Christian Seis.

• Robert Kerr (Warwick)

Title: Scaling of Navier-Stokes trefoil reconnection
Abstract: The reconnection of a trefoil vortex knot is examined numerically to determine how its helicity and two vorticity norms behave. During an initial phase, the helicity is remarkably preserved, as reported in recent experiments (Scheeler et al. 2014a). Equally unexpected is self-similar growth in the volume-integrated vorticity squared or enstrophy $Z$, growth where $\sqrt{\nu}Z(t_x)$ is independent of the viscosity at a configuration dependent time $t_x$ which will be interpreted as the end of first reconnection. By rescaling $t_x-t$ for times $t<t_x$ using viscosity dependent timescales $\delta t_\nu=T_c(\nu)-t_x$, one finds that the $1/\bigl(\sqrt{\nu}Z\bigr)^{1/2}=C(T_c(\nu)-t)/\delta t_\nu$ collapse onto one curve which is linear for $(T_c(\nu)-t)/\delta t_\nu<0.75$. Very small viscosities at early times are used to show that the Navier-Stokes $Z$ can, for a brief period, grow faster than the Euler $Z$ as the viscosity decreases.
This is observed even though $\|\omega\|_\infty$ is bounded by the Euler values for all $\nu$. Over the times
simulated, the velocity norm $L_3$ never changes more than the kinetic energy, which barely decreases until
very late times, $t>4t_x$. Furthermore, by $t_\epsilon\approx 2t_x$, a viscosity independent dissipation $\epsilon=\nu Z$ appears. Taken together, these results could be a new template whereby smooth solutions without singularities or roughness can generate the $\nu\rightarrow0$ dissipation anomaly (finite dissipation in a finite time) that is observed in all turbulent flows.

• Gabriel Koch (University of Sussex)

Title: Regularity criteria for the Navier-Stokes Cauchy problem in critical Besov spaces
Abstract: We consider the Cauchy problem for 3-D Navier-Stokes with data in any critical Besov space in which local existence has so far been established. If a singularity develops in finite time, we show (with I. Gallagher and F. Planchon) that the Besov norm must not be bounded. This extends a well-known result of Escauriaza-Seregin-Sverak for the critical Lebesgue space. Recently, G. Seregin has improved their result by showing that the Lebesgue norm must strongly tend to infinity at the blow-up time. We will also mention a work in progress with T. Barker on extending Seregin's result to the Besov setting.

• Igor Kukavica (University of Southern California)

Title: On the existence and uniqueness of solutions to a fluid-structure system
Abstract: We address the system of partial differential equations modeling motion of an elastic body inside an incompressible fluid. The fluid is modeled by the incompressible Navier-Stokes equations while the structure is represented by the damped wave equation with interior damping. We will review the local for large and global existence results for small data. The global existence result is obtained for small initial data in a suitable Sobolev space and is based on an exponential decay of solutions. The results are joint with M. Ignatova, I. Lasiecka, and A. Tuffaha.

• Piotr Mucha (Warsaw)

Title: Inhomogeneous Navier-Stokes system in exterior domains in critical functional framework.
Abstract: I plan to talk about a result for the incompressible Navier-Stokes system in an exterior domain. The main goal is to obtain regularity in space of type $L_1(0,T;\dot B^{s}_{p,1}(\Omega))$ in critical framework with respect to the nonlinearity. The main challenge is the result for the Stokes system. The chosen regularity fits very well to needs of the Lagrangian coordinates, then naturally we are able to consider the system with variable density. The talk will based on results from monograph: Critical functional framework and maximal regularity in action on systems of incompressible flows. Mem. Soc. Math. Fr. (N.S.) No. 143 (2015), joint with Raphael Danchin (Paris).

• Koji Ohkitani (Sheffield)

Title: The Navier-Stokes equations and Wiener path integrals: near-invariance under dynamic scaling in critical spaces
Abstract: Using probabilistic methods, we study the basic issues of the incompressible Navier-Stokes equations. We compare in detail the 3D Navier-Stokes equations written in the vector potentials and their dynamically-scaled counterpart, the Leray equations. On the basis of path integral formulations, we derive and discuss conditions for their global regularity.

• Benjamin Pooley (Warwick)

Title: A model for the Navier--Stokes equations in magnetization variables
Abstract: The magnetization variables formulation of the Navier--Stokes equations can be written as $w_t+(u\cdot\nabla)w +(\nabla u)^\top w-\nu\Delta w=0,\ u=\mathbb{P} w,$

where $\mathbb{P}$ denotes the Leray projection. We will review the equivalence between this system and the classical one, and introduce the following model system:

$w_t+(\mathbb{P} w\cdot\nabla)w +\frac{\nu}{2}\nabla |w|^2-\Delta w=0.$

This model admits a maximum principle and we will discuss a global existence and uniqueness result for this system in the Sobolev space $H^{1/2}(\mathbb{T}^3)$.

Title: Mild Navier-Stokes approximations by iteration of linear singular Volterra integral equations
Abstract: We approximate mild solutions to the Navier-Stokes initial-boundary value problem in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, during any compact interval $J$ of time. We use a variant of the classical Kato-Fujita, Giga-Miyakawa approximation scheme within a scale of Banach spaces embedded in some $L^r(\Omega), 1 < r < \infty$: At each iteration step we solve a linear singular Volterra integral equation. Observing monotonicity of bounds to the resolvent kernels we find again uniform convergence to a solution with error estimates in case of sufficiently short times in $J$, or on the whole of $J$ in case of sufficiently small datas (with respect to a suitable sup-norm). Beyond it
there results a stability statement concerning continuous solutions defined on $J$: The set of all initial values and exterior forces which admit continuous solutions on the whole of $J$ is open with respect to the sup-norm in question.

• Gregory Seregin (Oxford)

Title: Time decay for solutions to Stokes system with drift
Abstract: I am going to discuss behaviour of solutions to the Cauchy problem for the Stokes system with a drift as time goes to infinity.

• Roman Shvydkoy (University of Illinois at Chicago)

Title: Mechanisms for energy balance restoration in the Onsager (super)critical flows
Abstract: We will discuss several cases of solutions to equations of incompressible fluids which preserve the natural energy balance relations despite being Onsager critical or supercritical. These are the cases that exploit additional mechanisms such as symmetries coming from Hamiltonian structure, maximum principle, particular organization of singularity set, etc.

• Chuong Van Tran (St. Andrews)

Title: The pressure force in Navier--Stokes flows
Abstract: This talk analyses the pressure force in Navier--Stokes flows, with an emphasis on its apparent depletion of nonlinearity. Regularity criteria are derived and discussed.

• Luis Vega (Universidad del Pais Vasco & BCAM)

Title: Self-similar solutions of the binormal flow: transfer of energy
Abstract: Firstly I will review the construction of solutions of the binormal flow that develop a singularity in the shape of a corner which, nevertheless, can be continued after the singularity appears. Then, I will present a recent result about the discontinuity of some appropriate norm at the singularity time. Motivation comes from some numerical simulations on the evolution of regular polygons.