Related papers: On the Ricci-flat metric for the Navier-Stokes equ…
This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which…
General considerations on the Equivalence conjectures and a review of few mathematical results.
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…
We will consider the Navier-Stokes equation on a Riemannian manifold M with Ricci tensor bounded below, the involved Laplacian operator is De Rham-Hodge Laplacian. The novelty of this work is to introduce a family of connections which are…
Introduction: the Navier-Stokes equations are essential in fluid dynamics, describing the motion of fluids like liquids and gases. Solving these equations, especially in complex flows and high-Reynolds-number regimes, is a significant…
This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group.…
We study the two-dimensional stationary Navier-Stokes equations describing the flows around a rotating obstacle. The unique existence of solutions and their asymptotic behavior at spatial infinity are established when the rotation speed of…
We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the…
An algorithm for generating a class of closed form solutions to the Navier-Stokes equations is suggested, with examples. Of particular interest are those exact solutions that exhibit intermittency, tertiary Hopf bifurcations, flow reversal,…
We consider the two-dimensional Navier-Stokes equations subject to the Dirichlet boundary condition in a half plane for initial vorticity with finite measures. We study local well-posedness of the associated vorticity equations for measures…
In this paper, we study a special class of Finsler metrics, $(\alpha,\beta)$-metrics, defined by $F = \alpha \phi(\frac{\beta}{\alpha})$, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form. We find an equation that characterizes…
Exploring the possibility of describing a fluid flow via a time-reversible equation and its relevance for the fluctuations statistics in stationary turbulent (or laminar) incompressible Navier-Stokes flows.
The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise flat approximations of smooth manifolds is adapted to avoid an inherent…
We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat…
We put into light Ricci vector fields on $\mathbb R^2$ endowed with a diagonal metric.
We construct solutions to the Navier-Stokes equations on $\mathbf{R}^2$ with an arbitrary number of stagnation points which merge and split along trajectories that can be prescribed freely, up to a small deformation.
We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well…
Efficient formulae of Ricci tensor for an arbitrary diagonal metric are presented.
Ricci-flat metrics of the ultrahyperbolic signature which enjoy the l-conformal Galilei symmetry are constructed. They involve the AdS_2-metric in a way similar to the near horizon black hole geometries. The associated geodesic equations…
This is an expository paper which aims to give a simple proof of the existence of Ricci-flat metrics on certain K3 surfaces, as an illustration of general "glueing" techniques.