Numerical study of multiple periodic flow states in spherical Couette flow

The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method. For comparison with experiments of Nakabayashi et al. and Wimmer, a narrow gap and a medium gap with clearance ratio β=0.06 and 0.18 respectively are considered for the Reynolds number range covering the     

Fourier–Bessel theory on flow acoustics in inviscid shear pipeline fluid flow

Flow acoustics in pipeline is of considerable interest in both industrial application and scientific research. While well-known analytical solutions exist for stationary and uniform mean flow, only numerical solutions exist for shear mean flow. Based on potential theory, a general mathematical formulation of flow acoustics in inviscid fluid with shear mean flow is deduced, resulting in a set of two second-order differential equations. According to Fourier–Bessel theory which is orthogonal and complete in Lebesgue Space, a solution is proposed to transform the differential equations to linear homogeneous algebraic equations. Consequently, the axial wave number is numerically calculated due to the existence condition of non-trivial solution to homogeneous linear algebraic equations, leading to the vanishment of the corresponding determinant. Based on the proposed method, wave propagation in laminar and turbulent flow is numerically analyzed.     

Hyperbolic Khler-Ricci flow

In this paper, the author has considered the hyperbolic Khler-Ricci flow introduced by Kong and Liu, that is, the hyperbolic version of the famous Khler-Ricci flow. The author has explained the derivation of the equation and calculated the evolutions of various quantities associated with the equation including the curvatures. Particularly on Calabi-Yau manifolds, the equation can be simplifled to a scalar hyperbolic Monge-Ampère equation which is the hyperbolic version of the corresponding one in Khler-Ricci flow.     

Immortal solution of the Ricci flow

For any complete noncompact Kahler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow.     

Hele–Shaw flow on hyperbolic surfaces

Consider a complete simply connected hyperbolic surface. The classical Hadamard theorem asserts that at each point of the surface, the exponential mapping from the tangent plane to the surface defines a global diffeomorphism. This can be interpreted as a statement relating the metric flow on the tangent plane with that of the surface. We find an analogue of Hadamard's theorem with metric flow replaced by Hele–Shaw flow, which models the injection of (two-dimensional) fluid into the surface. The Hele–Shaw flow domains are characterized implicitly by a mean value property on harmonic functions.     

On flow of a Navier–Stokes fluid in curved pipes. Part I: Steady flow

We consider the steady, fully developed motion of a Navier–Stokes fluid in a curved pipe of cross-section $D$ under a given axial pressure gradient $G$. We show that, if $G$ is constant, this problem has a smooth steady solution, for arbitrary values of the Dean’s number $\kappa$, for $D$ of arbitrary shape and for any curvature ratio $\delta$ of the pipe. This solution is also unique for $\kappa$ sufficiently small. Moreover, we prove that the solution is unidirectional (no secondary motion) if and only if $\kappa =0$. Finally, we show the same properties for the approximations to the Navier–Stokes equations called “Dean’s equations” and provide a rigorous way in which solutions to the full Navier–Stokes equations approach those to this approximation in the limit of $\delta \to 0$.     

Ricci flow on Kähler manifolds

In this Note, we announce the result that if M is a Kähler–Einstein manifold with positive scalar curvature, if the initial metric has nonnegative bisectional curvature, and the curvature is positive somewhere, then the Kähler–Ricci flow converges to a Kähler–Einstein metric with constant bisectional curvature.     

A modified Kähler–Ricci flow

In this note, we study a Kähler–Ricci flow modified from the classic version. In the non-degenerate case, strong convergence at infinite time is achieved. The main focus should be on degenerate case, where some partial results are presented.     

A 2 × 2 hyperbolic traffic flow model

A new traffic flow model is presented. It accounts for various qualitative features of the evolution of the density and speed of cars along a crowded road. The model consists of a 2 × 2 system of nonlinear hyperbolic conservation laws generating a Cauchy problem which is well posed for all reasonable initil data. A similar result can be proved for the initial boundary value problem. The presence of a speed limit is also considered.     

Exponential mixing for the Teichmüller flow

We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the $SL(2,\mathbb{R})We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of H?lder observables. A geometric consequence is that the action in the moduli space has a spectral gap.     

Decent flow methods for inverse Sturm–Liouville problem

In this paper, we propose three numerical methods for the inverse Sturm–Liouville operator in impedance form. We use a finite difference method to discretize the Sturm–Liouville operator and expand the impedance function with some basis functions. The correction technique is discussed. By solving an un-weighted least squares problem, we find an approximation to the impedance function. Numerical experiments are presented to show the accuracy and stability of the numerical methods.     

Geodesic flow, connecting orbits and almost full foliation

We study in this article a special dynamical behavior of geodesic flow on T2. Our example shows that there is an area-preserving monotone twist map for which all minimal periodic orbits can be connected, and at the same time for a certain rational rotation number the minimal set is almost an invariant curve.     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

Morse homology for the Yang–Mills gradient flow

We use the Yang–Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse chain complex. The chain groups are generated by Yang–Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang–Mills gradient flow lines that converge asymptotically to Yang–Mills connections.     

A note on K?hler–Ricci flow

Let g(t) with ${t\in [0,T)}$ be a complete solution to the K?hler–Ricci flow: ${\frac{d}{dt}g_{i\bar j}=-R_{i\bar j}}$ where T may be ∞. In this article, we show that the curvature of g(t) is uniformly bounded if the solution g(t) is uniformly equivalent. This result is stronger than the main result in ?e?um (Am J Math 127(6):1315–1324, 2005) within the category of K?hler–Ricci flow.     

Regularity of the Kähler–Ricci flow

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.     

Local Schr?dinger flow into K?hler manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schr?dinger flow for maps from a compact Riemannian manifold into a complete K?hler manifold, or from a Euclidean space R^m into a compact K?hler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.     

Ancient shrinking spherical interfaces in the Allen–Cahn flow

We consider the parabolic Allen–Cahn equation in ${\mathbb{R}}^n$, $n\ge 2$,

$u_t=\mathrm{\Delta }u+(1?u^2)u\text{ in }{\mathbb{R}}^n\times (?\infty ,0].$

We construct an ancient radially symmetric solution $u(x,t)$ with any given number k of transition layers between ?1 and +1. At main order they consist of k time-traveling copies of w with spherical interfaces distant $O(\log ?|t|)$ one to each other as $t\to ?\infty$. These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: $|x|=\sqrt{?2(n?1)t}$. More precisely, if $w(s)$ denotes the heteroclinic 1-dimensional solution of $w^{″}+(1?w^2)w=0$$w(\pm \infty )=\pm 1$ given by $w(s)=\tanh ?\left(\frac{s}{\sqrt{2}}\right)$ we have

$u(x,t)\approx \sum _{j=1}^k{(?1)}^{j?1}w(|x|?{\rho }_j(t))?\frac{1}{2}(1+{(?1)}^k)\text{ as }t\to ?\infty$

where

${\rho }_j(t)=\sqrt{?2(n?1)t}+\frac{1}{\sqrt{2}}(j?\frac{k+1}{2})\log ?\left(\frac{|t|}{\log ?|t|}\right)+O(1),j=1,\dots ,k.$