An important property of the 3D Euler equations is that the topology of the vortex structures of the fluid does not change in time as long as the solutions do not develop any singularities. To put it differently, the set of (say) vortex tubes and vortex lines of the fluid at time t is diffeomorphic to that of the initial vorticity, provided that the solution remains smooth up to this time. Of course, numerical simulations and experiments with real fluids have shown that the situation is completely different in the case of viscous fluids. In this talk we will show how vortex tubes and vortex lines, of arbitrarily complex topologies, are created and destroyed in smooth solutions to the 3D Navier-Stokes equations. This is joint work with Renato Luca and Daniel Peralta-Salas.Blowup for model equations for fluid mechanics, Vu Hoang (Rice University)
In this talk, I discuss recent results for various model equations of fluid mechanics. My focus lies on achieving finite-time blowup in a controlled manner using the hyperbolic flow scenario. A central theme is to uncover the blowup mechanics by considering simplified velocity fields and vorticity stretching mechanisms.
This talk is based on joint work with T. Do, M. Radosz and X. XuGlobal existence for fluid-structure models, Mihaela Ignatova (Princeton University)
We address a fluid-structure system coupling the incompressible Navier-Stokes equation and a linear elasticity equation with interior damping. The interaction takes place at a common interface and it is described by the transmission boundary conditions matching the velocities and the stress forces at the interface. We prove the global existence and exponential decay of smooth solutions for small initial data. This is a joint work with Igor Kukavica, Irena Lasiecka, and Amjad Tuffaha.Anomalous Diffusion and Intermediate time Homogenization for cellular flows, Gautam Iyer (Carnegie-Mellon University)
Cellular flows arise in various contexts, most notably as a as a two dimensional model for heat transport in Bernard convection cells. Our interest is to study the behaviour of a passive scalar diffusing in the presence of a cellular flow. Well known homogenization results show that the long time behaviour can be modelled by the heat equation with an enhanced diffusion coefficient. In stark contrast, we show that on intermediate time the effective behaviour is instead governed by a time fractional equation. The proof is probabilistic, using the framework of Freidlin-Wentzel. This is joint work with M. Hairer, L. Koralov, A. Novikov and Z. Pajor-Gyulai.On steady solutions for the 2D Euler equation, Anton Izosimov (University of Toronto)
The motion of an ideal incompressible fluid on a 2D surface is described by the Euler equation, which can be regarded as a Hamiltonian system on coadjoint orbits of the symplectic diffeomorphisms group. Using a combinatorial description of these orbits in terms of graphs with some additional structures, we give a characterization of coadjoint orbitswhich may admit steady solutions of the Euler equation (steady fluid flows).
This is a joint work with B.Khesin.Geometric models of the 3D Euler equation, Steven Preston (CUNY)
I will give a survey of the geometric properties of the Euler equations from the Arnold perspective, viewed as geodesics on the infinite-dimensional group of volume-preserving diffeomorphisms, in particular the geometric differences between 2D and 3D Euler. Then I will discuss geometric properties of two model equations of 3D Euler: the surface quasigeostrophic equation in 2D (which was recently shown to fit into the Euler-Arnold framework by Terry Tao) and the Wunsch equation in 1D (which is related to the models of De Gregorio, Hou-Luo, and Constantin-Lax-Majda). This is joint work with my graduate student Pearce Washabaugh.Global smooth solutions for the inviscid SQG equations, Javier Gomez-Serrano (Princeton University)
Motivated by our previous results of global existence for active scalars in the patch setting, we are able to construct the first nontrivial family of global smooth solutions for the surface quasi-geostrophic (SQG) equations. These solutions rotate with uniform angular velocity both in time and space. Joint work with Angel Castro and Diego Cordoba.Homogeneous solutions to the incompressible Euler equation, Roman Shvydkoy (UI-Chicago)
In this talk we describe recent results on classification and rigidity properties of stationary homogeneous solutions to the 3D and 2D Euler equations. The problem is motivated by several connections with other questions such as anomalous dissipation, intermittency, and self-similar blowup. In 2D the problem also arises in isometric immersions theory and optimal transport. A full classification of two dimensional solutions will be given. In 3D we reveal several new classes of solutions and prove their rigidity properties. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function; and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler equation on the sphere. We further discuss the case when homogeneity corresponds to the Onsager-critical state. We will show that anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme 0-dimensional intermittencies.Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima-Charney-Obukhov Model, Edriss Titi (Texas A&M University and Weizmann Institute)
The 3D inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The same model is known as the Charney- Obukhov model for stratified ocean dynamics, and also appears in literature as a simplified reduced Rayleigh-B ́enard convection model. The mathematical analysis of the Hasegawa-Mima and of the Charney-Obukhov equations is chal- lenging due to the their resemblance with the Euler equations. In this talk, we introduce and show the global regularity of a model which is inspired by the inviscid Hasegawa-Mima and Charney-Obukhov models, named a pseudo- Hasegawa-Mima model. The introduced model is easier to investigate ana- lytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. To establish our global regularity result we implement a new logarithmic inequality, generalizing the Brezis-Gallouet-Berzis-Wainger inequalities. (This is a joint work with C. Cao and A. Farhat.)On the vanishing viscosity limit for the Navier-Stokes equations, Vlad Vicol (Princeton University)
We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the Navier-Stokes solutions remain bounded in $L^2_t L^\infty_x$ independently of the kinematic viscosity, and if they are equicontinuous at $x_2 = 0$. These conditions imply that there is no boundary layer separation: the Lagrangian paths originating in a boundary layer, stay in a proportional boundary layer during the time interval considered.
We then give a proof of the conjecture of vanDommelen and Shen (1980) which predicts the finite time blowup of the displacement thickness in the Prandtl boundary layer equations. This shows that the Prandtl layer exhibits dynamic separation in finite time.
This talk is based on joint work with Peter Constantin, Tarek Elgindi, Mihaela Ignatova; and Igor Kukavica, Fei Wang.The two-dimensional Boussinesq equations with partial or fractional dissipation, Jiahong Wu (Oklahoma State University)
The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. In addition, they play an important role in the study of Rayleigh-Benard convection. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. The global regularity problem on the 2D Boussinesq equations with partial or fractional dissipation has attracted considerable attention in the last few years. This talk presents recent developments in this direction. In particular, we detail the global regularity result on the 2D Boussinesq equations with vertical dissipation as well as some recent work on the 2D Boussinesq equations with general critical dissipation. If time permits, we will also briefly discuss the regularity problem on the partially dissipated Boussinesq equations in a bounded domain.Finite time singularity of a vortex patch model in the half plane, Yao Yao (Georgia Tech)
The question of global regularity v.s. finite time blow-up remains open for many fluid equations. In this talk, I will discuss a family of equations which interpolate between the 2D Euler equation and the surface quasi-geostrophic (SQG) equation. We focus on the patch dynamics for this family of equation in the half-plane, and obtain the following results: For the 2D Euler patch model, the patches remain globally regular even if they initially touch the boundary of the half-plane; whereas for the family of equations that are slightly more singular than the 2D Euler equation, the patches can develop a finite-time singularity. This talk is based on a joint work with A. Kiselev, L. Ryzhik and A. Zlatos.