Related papers: Special solutions for Ricci flow equation in 2D us…
We give an unified framework to solve rough differential equations. Based on flows, our approach unifies the former ones developed by Davie, Friz-Victoir and Bailleul. The main idea is to build a flow from the iterated product of an almost…
Ricci and sectional curvatures of twisted flux tubes in Riemannian manifold are computed to investigate the stability of the tubes. The geodesic equations are used to show that in the case of thick tubes, the curvature of planar (Frenet…
We introduce a classification conjecture for $\kappa$-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of $\mathbb{Z}_2^2\times \mathrm{O}_3$-symmetric…
An iterative scheme is presented to solve analytically the relativistic fluid dynamics equations. The scheme is applied to longitudinal expansion, transversal symmetric and transversal asymmetric (triaxial) expansion as well. Within this…
We study a first variation formula for the eigenvalues of the Laplacian evolving under the Ricci flow in a simple example of a noncommutative matrix geometry, namely a finite dimensional representation of a noncommutative torus. In order to…
As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently…
The widely studied benchmark problem, 2-D driven cavity flow problem is discussed in details in terms of physical and mathematical and also numerical aspects. A very brief literature survey on studies on the driven cavity flow is given.…
I summarize our recent work towards finding and utilizing analytic solutions of relativistic hydrodynamic. In the first part I discuss various exact solutions of the second-order conformal hydrodynamics. In the second part I compute flow…
Numerical calculations of the 2-D steady incompressible driven cavity flow are presented. The Navier-Stokes equations in streamfunction and vorticity formulation are solved numerically using a fine uniform grid mesh of 601x601. The steady…
Partially invariant solution to (2+1)D shallow water equation is constructed and investigated. The solution describes an extension of a stripe, bounded by linear source and drain of fluid. Realizations of smooth flow and of hydraulic jump…
In this paper, we find various analytic (1+3)D solutions to relativistic ideal hydrodynamic equations based on embedding of known low-dimensional scaling solutions. We first study a class of flows with 2D Hubble Embedding, for which a…
We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.
We decribe and announce some results (joint with G. Besson, L. Bessieres, M. Boileau and J.Porti) about the geometry and topology of 3-manifolds. Most of the article is primarily intended as an introduction for nonexperts to geometrization…
Ricci flow spacetimes were introduced by Kleiner & Lott as a way to describe Ricci flow through singularities, and have since been used elsewhere in the literature, prompting the question of their rigidity. In $(2+1)$-dimensions, we show…
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1)…
Numerical solutions of 2-D steady incompressible flow inside a triangular cavity are presented. For the purpose of comparing our results with several different triangular cavity studies with different triangle geometries, a general triangle…
We prove results relating the theory of optimal transport and generalized Ricci flow. We define an adapted cost functional for measures using a solution of the associated dilaton flow. This determines a formal notion of geodesics in the…
We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.
We derive Hamiltonian flow equations giving the evolution of the Lipkin Hamiltonian to a diagonal form using continuous unitary transformations. To close the system of flow equations, we present two different schemes. First we linearize an…
In this paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2.