Schedule for: 23w5020 - Infinite Dimensional Geometry and Fluids

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

An inclusive framework for joined Hamiltonian and dissipative dynamical systems, which preserve energy and produce entropy, is given. The dissipative dynamics of the framework is based on the metriplectic 4-bracket, a quantity like the Poisson bracket defined on phase space functions, but unlike the Poisson bracket has four slots with symmetries and properties motivated by Riemannian curvature. Metriplectic 4-bracket dynamics is generating using two generators, the Hamiltonian and the entropy, with the entropy being a Casimir of the Hamiltonian part of the system. The formalism ensures thermodynamic consistency, and includes all known previous binary bracket theories for dissipation or relaxation as special cases. Rich geometrical significance of the formalism and methods for constructing metriplectic 4-brackets, including Kulkarni-Nomizu and Lie algebra based methods, are explored. Realizations on Poisson manifolds with the Fernandes-Koszul connection are also explored. Many physical examples of both finite and infinite dimensions will be discussed.

In this talk, we give show that there is no local minimum to the H^{-1} norm in the Wasserstein geometry. On closed Riemannian manifolds, we reduce the question of global convergence with a linear rate of convergence to a regularity property of the solution. We then discuss open questions related to global regularity.

We present a topological analysis of the vorticity formulation of the incompressible Euler equation. In particular, we elucidate the equations of motion for the often-omitted cohomology component of the velocity on non-simply-connected domains. These equations have nontrivial coupling with the vorticity, which is crucial for characterizing correct vortex motions with presence tunnels or islands in the domain. The dynamics of fluid’s cohomology is also associated with new conservation laws as Casimir invariants. These conservation laws extend the symplectic leaves from the classical Poisson space of vorticity to a larger Poisson space including the cohomology information. Additional results include new analytical solutions of the Euler equations that are unsteady; and the first general vortex-streamfunction-based numerical method on curved surfaces with arbitrary genus and boundaries that is consistent with the Euler equation.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In a more general version of the ideal fluid dual pair, one that involves the group of diffeomorphisms preserving a weighted nonlinear flag, symplectic reduction for the right leg yields coadjoint orbits of the Hamiltonian group, consisting of weighted nonlinear flags. This is joint work with Stefan Haller and Cornelia Vizman.

It is well-known that a pair of point vortices of opposite circulations (a dipole) on the plane moves in a straight line. More generally, it was conjectured by Kimura and later proved by Boatto and Koiller that on a closed surface without boundary a "singular" dipole (limit of considing vortices) traces a geodesic. Here we consider the motion of singular dipoles on domains with boundaries and show that they move according to a "pensive billiard system". Specifically, the singular dipole moves on the geodesic until it hits the boundary, at which point it splits into its two monopole components which move along the boundary with constant speeds in opposite directions. The ratio of speeds is related to the angle of geodesic hitting the boundary. If the boundary is closed, monopoles can meet and may merge into dipoles which reenter the domain. Thus the trajectory of a dipole appears to hit the boundary and reflect back into the domain from a different location some time later.

I present an implicit representation of space curves. Instead of explicitly parametrizing curves, I represent them implicitly with a co-dimension 2 variant of the so-called Clebsch variables. I then explain our ongoing challenge about a question "Does such implicit curves admit a symplectic structure like the celebrated Marsden-Weinstein form for explicit curves?"

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The SQG family are active scalar equations that interpolate between the 2D Euler and the standard surface quasi-geostrophic equations. Not surprisingly, they share a lot of analytic (and geometric) properties with the latter though to exhibit them is technically somewhat more complicated. I will describe how the corresponding solution maps of the SQG family are at best continuous. This is joint work with Truong Vu.

We describe a new method for proving blowup of certain Euler-Arnold equations, partial differential equations which represent geodesics on groups of diffeomorphisms under right-invariant metrics. It is based on using momentum conservation to treat the equation as a first-order ODE on a Banach space, then using proving that solutions breakdown based on a comparison theorem using the known exact solution $f$ of the classical Liouville equation. Applications are given for the right-invariant $H^2$ Sobolev metric on the group of diffeomorphisms of $\mathbb{R}^n$, where we show that solutions can break down in finite time if $n\ge 3$.

We will discuss aspects of the long term dynamics of 2d perfect fluids. As an application of a certain stability of twisting for general hamiltonian flows, we will show generic loss of smoothness near stable steady states, the existence of many wandering neighborhoods, aging of the Lagrangian flow, along with other examples of complex behavior such as indefinite perimeter growth for special vortex patches.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Half Lie groups exist only in infinite dimensions: They are smooth manifolds and topological groups such that right translations are smooth. Main examples are Sobolev $H^r$-diffeomorphism groups of compact manifolds, or $C^k$-diffeomorphism groups, or semidirect products of a Lie group with kernel an infinite dimensional representation space (investigated by Marquis and Neeb). Here, we investigate mainly Banach half Lie groups, the groups of their $C^k$-elements, extensions, and right invariant strong Riemannian metrics on them: Here surprisingly the full Hopf Rinov theorem holds which is wrong in general even for Hilbert manifolds.

We consider the group $D=SDiff(M)$ of volume preserving diffeomorphisms of a bounded domain $M\subset\R^n$. Every map $f\in D$ can be tautologically regarded as a map $f:M\to \R^n$; if we consider these maps in the Sobolev $H^s$ metric, then we can regard $D$ as a subset of $X=H^s(M,\R^n)$. The geometric properties of the set $D\subset \R^n$ depend a great deal on the Sobolev exponent $s$. We consider two cases. (1) If $s>n/2+2$ then $D$ is a smooth submanifold of $X$. Moreover, $D$ is a "Quasiruled manifold", i.e. it can be uniformly approximated by the "Ruled manifolds", which are the unions of a finite-dimebsional family of affine planes in $X$. Such manifolds form a category $QL(X)$; the morphisms in this category are "Fredholm Quasiruled maps" (FQR-maps). The main result here is the following Theorem: Let $f_t\in D$ be the family of flow maps defined by the Lagrange-Euler equations of theideal incompressible fluid. Then for any $t$, $f_t: D\to D$ is an FQR-map wherever it is defined; for $n=2$ the maps $f_t$ are defined all over $D$ for any $t$. This theorem justifies the use of topological methods in the study of the fluid kinematics. (2) If $s=0$ (i.e. $X=L^2(M,\R^n)$), then $D\subset X$ is not a smooth manifold; it is quite a singular set in $X$. If $n=2$, the intrinsic metric of $D$ is unrelated to the metric induced from $X$ (in particular, $D$ is not metrically connected); for $n>2$ $D$ is metrically connected but the relation between the intrinsic and the induced metrics looks like for the the Carnot-Caratheodori spaces. The set $D$ does not have a true tangent space $T_f D$ at a point $f\in D$, but we are able to define its substitute $Y_f$; using this structure, we can define a class of dissipative weak solutuions for the Euler equations for $n>2$ which may have a physical sense.

Arnold showed that solutions to the hydrodynamical Euler equation can be interpreted as geodesics on the group of volume-preserving diffeomorphisms. Motivated by insolvability of the two-point problem for the Euler equation (i.e. non-existence of minimizing geodesics between certain pairs of fluid configurations), Brenier and Shnirelman considered its relaxed version, leading to so-called generalized flows. In the talk I will describe the geometry behind such flows. This is based on joint work with Boris Khesin.

The generalized SQG family of equations interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation; a well known model for the three dimensional Euler equations. These equations can be realised as geodesic equations on groups of diffeomorphisms. I will present some results pertaining to Fredholmness of the corresponding exponential maps, as well as the distribution of conjugate points. This is joint work with Martin Bauer, Gerard Misiołek and Stephen Preston.

We will study magnetic deformations of the Hunter-Saxton system, in the sense of magnetic geodesic flows. We represent this system as a Hamiltonian flow on an infinite dimensional Lie group and use this to study blow ups and construct weak solutions of this system of partial differential equations. Furthermore we will use the global weak magnetic flow on the infinite dimensional Lie group to prove that any two points in there can be connected by a magnetic geodesic as long as the strength of the magnetic field is less then Mañé's critical value.

Joint work with B. Khesin and K. Modin. A simple model capturing the key features of unbalanced optimal transport will be explored with an emphasis on a Riemannian submersion of the diffeomorphism group to the space of volume forms of arbitrary mass, as well as a finite-dimensional analogue of this submersion.

SDEs on manifolds are formulated using Schwartz operators in the context of second order differential geometry. Conversely, most results in second order differential geometry have a probabilistic interpretation. Within this framework, Stratonovich differential equations provide a counterpart to first order differential geometry and ODEs, and their reduction and reconstruction have been studied by Lazaro-Camí and Ortega and others. We aim to generalise these ideas to Schwartz operators.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The work of Etnyre and Ghrist has unveiled a profound correspondence between Reeb vector fields and Beltrami fields, stationary solutions to Euler's equations. This correspondence enables the transfer of diverse geometric features ( existence of periodic orbits, universality, etc.), which are reflected from the domain of Reeb vector fields to the intricate landscape of fluid dynamics as a mirror. Other groundbreaking insights by Ginzburg and Khesin associate a symplectic structure defined almost-everywhere to four-dimensional steady flows. In this talk, we will present some applications of this correspondence, with a particular focus on three-dimensional scenarios and we will unveil new mirrors when the geometrical structure under consideration is not contact. Time permitting, I will end up my talk with open questions concerning the theory of confoliations. This is joint work in progress with Ángel González Prieto and Daniel Peralta-Salas.

Magnetic relaxation is a mechanism that aims to obtain magnetohydrostatic (MHS) equilibria as long-time limits of a topology-preserving evolution equation, hopefully easier to analyze that the original ideal MHD equations. My goal in this talk is to present a new theorem, in collaboration with A. Enciso, on generic obstructions for a divergence-free vector field to be topologically equivalent to some MHS equilibrium. Specifically, I will show that for any axisymmetric toroidal domain there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any MHS equilibrium. Each vector field in this set is Morse-Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain.

A variational geometric setting has recently been proposed for nonequilibrium thermodynamics, which extends the Hamilton principle and the geometric formulation of classical mechanics to include irreversible phenomena. This setting can be developed for fluid dynamics, thereby extending the geometric formulation of fluids on diffeomorphism groups to include irreversible processes such as viscosity and heat conduction. We shall review this geometric framework and show how it can be discretized to yield structure preserving and thermodynamically consistent finite element schemes. We shall focus on the Navier-Stokes-Fourier equations with several types of thermal boundary conditions, and show how our results improve existing numerical methods, especially concerning the laws of thermodynamics.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.

ingularity formation in incompressible fluids is known to be very challenging. One of the main difficulties is controlling the nonlocal terms in the equations due to the incompressible condition. In this talk, we will discuss some sharp functional inequalities based on optimal transport that provide effective estimates of the nonlocal terms in suitable functional spaces in the analysis of singularity formation. Time permitting, we will briefly discuss another challenge in this field. Some geometry ideas may shed light on it.

In this talk, we consider the problem of identifying the long-time behaviour of a 2D ideal fluid, governed by the Euler equations. The predictions made by statistical mechanics have found mixed success in experiments and numerical simulations. Different methods have shown that it is likely that unsteadiness persists indefinitely, therefore it is not clear in which sense the limit state of a fluid should be identified. Nevertheless, in the last decades, Shnirelman has proposed a systematic way to determine weak-* limits for the vorticity field, when t → ∞. Here, we focus on the long-time behaviour of a finite dimensional analogue of the 2D Euler equations, developed by Zeitlin from the 90s. Quite recently, the results obtained via numerical simulations of the Euler–Zeitlin equations have received interest from a purely analytic perspective, as they qualitatively reproduce some of the features predicted by the theory of Shnirelman. Thanks to the structure of the Euler–Zeitlin equations, we are able to identify large and small scale structures in the long-time limit. Passing to the infinite dimensional limit of the Zeitlin model, the large scale structures identified can be viewed as suitable weak-* limits for the vorticity field, when t → ∞. Even if a complete picture in how the qualitative behaviour of the Euler–Zeitlin equations is linked to the infinite dimensional case is still missing, we show that our numerical experiments may help in identifying non-trivial limits for the vorticity field within the theory of Shnirelman.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will outline some results on the study of transfer of energy for solutions to the periodic 2D (torus domain) cubic defocusing nonlinear Schrodinger equation. In particular I will focus on the differences of the dynamics of solutions in the rational versus irrational torus.

The Muskat equation describes the interface of two liquids in a porous medium. We will show that if a solution to the Muskat problem in the case of same viscosity and different densities is sufficiently smooth, then it must be analytic except at the points where a turnover of the fluids happens. We will also show analyticity in a region that degenerates at the turnover points provided some additional conditions are satisfied.

We develop a framework for studying quasi-periodic maps and diffeomorphisms on $\mathbb{R}^n$. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on $\mathbb{R}^n$. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Hypersemitoric systems are two degree of freedom integrable Hamiltonian systems on 4-dimensional compact symplectic manifolds with possibly mild degeneracies where one of the integrals gives rise to an effective Hamiltonian $S^1$-action. We will give an update on recent developments like important new examples, links with Hamiltonian $S^1$-actions, bifurcation theory, symplectic features like (non)displaceability of fibers, and steps towards a symplectic classification.

TBA

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.