arXiv research

Study how large-scale flows align small-scale vortices in 3D Euler equations.

There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimension…

Existence of a conjugate point proven on 3D ellipsoid.

Researchers study surface area functionals in CR manifolds, deducing equations for various cases.

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we…

The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The local nature of this phenomenon suggests the appearance of singularity in nearb…

In 3D, conformal geodesics are variational.

Classifies 3D F-manifolds with or without Euler fields.

The paper proves a new discrete Laplacian for 3D meshes and shows its superiority over primal construction.

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…

In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this t…

Extends Zeitlin's model to 3-D axisymmetric Euler equations.

Three distinct 3D components found in local extremal Kähler metrics.

The paper disproves a conjecture about 3D manifolds using even lattice points.

Euler derived elastica equation using modern mathematical concepts.

Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.

In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming fro…

We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorp…

New solutions to 3D integrability equations using quantum cluster algebras.

Study on stability in discretized hydrodynamics model.

In this paper, we will generalize the Bott-Virasoro group, applying the concept of the connection cochain, and derive the Euler equations corresponding to the generalized Bott-Virasoro group. We will show the relationships between the new Euler equations and the old ones. Moreover, we will study the geodesic equation c…

New equations for Cosserat media motions derived from bundle automorphisms.

Study of 3d-3d correspondence involving $q$-Weyl algebra and 3d-index.

We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution (in the stochastic sense) in spaces…

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

This paper extends invariant Euler-Lagrange equations to higher dimensions and groups.

We develop an integral geometry of stationary Euler equations defining some function $w$ on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution $v$ of the system, and we deduce a linear differential equation for $w$. We prove also that the purported annulation…

We present a contact transformation of the generalized Hunter--Saxton equation to the Euler--Poisson equation with special values of the Ovsiannikov invariants. We also find the general solution for the generalized Hunter--Saxton equation.

Geometric framework for dissipative systems on Lie algebroids.

EuLearn creates diverse 3D topological datasets for machine learning.

The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.

Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and…

Paper derives invariantised Euler-Lagrange equations for Herglotz problems.

Study bends 2D surfaces in 3D space using special equations.

The 3D-index connects to Turaev-Viro invariant and knot periods.

Geometric mechanics approach to constrained and floating multibody systems using Hamel's equations.

In this paper we show that there are applications that transform the movement of a pendulum into movements in $\mathbb{R}^3$. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in…

Toolbox for stochastic Euler equations using Ebin-Marsden theory.

Develops higher-order Euler-Poincaré field equations for principal G-bundles.

Optical interpretation of Euler's angle problem for caustics of light rays.

Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariants with the modern tools …

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riema…

Develops methods for constructing exact, non-stationary solutions to Euler equations.

We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.

We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets …

We prove a version of the variational Euler-Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.

Study of closed real plane curves with hyperelliptic genus three solutions.

In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to zero, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler …