Searcharxiv⌕ Search

arXiv subjects

Yuxiang Li

Publications and source records attributed to Yuxiang Li.

At least 55 records · Page 3Linked to original sources
arXiv2021

$W^{1,p}$-metrics and conformal metrics with $L^{n/2}$-bounded scalar curvature

A $W^{1,p}$-metric on an $n$-dimensional closed Riemannian manifold naturally induces a distance function, provided $p$ is sufficiently close to $n$. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the corresponding distance functions $d_{g_k}$ subconverge to a limit distance function $d$, which satisfies $d\le d_g$. As an application, we show that the above convergence result applies to a sequence of conformal metrics with $L^{n/2}$-bounded scalar curvatures, under certain geometric assumptions. In particular, in this special setting, the limit distance function $d$ actually coincides with $d_{g}$.

math.DG↗
arXiv2020

Minimizers of nonlocal interaction functional with exogenous potential

The purpose of this paper is to consider the minimization problem of the following nonlocal interaction functional \begin{equation*} E[ρ]=\frac{1}{2}\int_{\mathbb{R}^N} \int_{\mathbb{R}^N}K(x-y)ρ(x)ρ(y)dxdy+\int_{\mathbb{R}^N}F(x)ρ(x)dx. \end{equation*} The kernel $K(x)=\frac{1}{q}|x|^q-\frac{1}{p}|x|^p$ is an endogenous potential, where $q>p>-N$. The exogenous potential $F$ is a nonnegative continuous function and satisfies $F(x)\to +\infty$ as $|x|\to +\infty$. The existence of minimizers are established based on the concentration compactness principle. Especially, for $F(x)=β|x|^2(β>0)$ and $K(x)=\frac{1}{2}|x|^2-\frac{1}{2-N}|x|^{2-N}$($N>2$), the global minimizer is given explicitly by the method of calculus of variation.

math.AP↗
arXiv2019

Metrics On A Surface With Bounded Total Curvature

Let $g=e^{2u}(dx^2+dy^2)$ be a conformal metric defined on the unit disk of $\mathbf{C}$. We give an estimate of $\|\nabla u\|_{L^{2,\infty}(D_\frac{1}{2})}$ when $\|K(g)\|_{L^1}$ is small and $\frac{μ(B_r^g(z),g)}{πr^2}<Λ$ for any $r$ and $z\in D_\frac{3}{4}$. Then we will use this estimate to study the Gromov-Hausdorff convergence of a conformal metric sequence with bounded $\|K\|_{L^1}$ and give some applications.

math.DG↗
arXiv2018

Global existence and boundedness in a chemotaxis-Stokes system with slow $p$-Laplacian diffusion

This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion \begin{equation}\nonumber \left\{ \begin{aligned} &n_t+u\cdot\nabla n=\nabla\cdot\left(|\nabla n|^{p-2}\nabla n\right)-\nabla\cdot(n\nabla c), &x\inΩ,\ t>0,\ \ &c_t+u\cdot\nabla c=Δc-nc,&x\inΩ,\ t>0,\ \ &u_t=Δu+\nabla P+n\nablaϕ,&x\inΩ,\ t>0,\ \ &\nabla\cdot u=0, &x\inΩ,\ t>0,\ \ \end{aligned} \right. \end{equation} where $ϕ\in W^{2,\infty}(Ω)$ is the gravitational potential. It is proved that global bounded weak solutions exist whenever $p>\frac{23}{11}$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0\geq 0$ and $c_0\geq 0$.

math.AP↗
arXiv2018

Renormalized solutions to a chemotaxis system with consumption of chemoattractant

This paper investigates a high-dimensional chemotaxis system with consumption of chemoattractant \begin{eqnarray*} \left\{\begin{array}{l} u_t=Δu-\nabla\cdot(u\nabla v), v_t=Δv-uv, \end{array}\right. \end{eqnarray*} under homogeneous boundary conditions of Neumann type, in a bounded convex domain $Ω\subset\mathbb{R}^n~(n\geq4)$ with smooth boundary. It is proved that if initial data satisfy $u_0\in C^0(\overlineΩ)$ and $v_0\in W^{1,q}(Ω)$ for some $q>n$, the model possesses at least one global renormalized solution.

math.AP↗
arXiv2018

Global generalized solutions to a nonlinear Keller-Segel equation with singular sensitivity

We consider the chemotaxis system \begin{eqnarray*} \begin{cases} \begin{array}{lll} \medskip u_t =Δu^m - \nabla(\frac{u}{v}\nabla v),&{} x\inΩ,\ t>0, \medskip v_t =Δv -uv,&{}x\inΩ,\ t>0, \medskip \frac{\partial u}{\partial ν}=\frac{\partial v}{\partialν}=0,&{}x\in\partialΩ,\ t>0, \medskip u(x,0)=u_0(x),\ v(x,0)=v_0(x), &{}x\inΩ, \end{array} \end{cases} \end{eqnarray*} in a smooth bounded domain $Ω\subset \mathbb{R}^n$, $n\geq2$. In this work it is shown that for all reasonably regular initial data $u_0\geq0$ and $v_0>0$, the corresponding Neumann initial-boundary value problem possesses a global generalized solution provided that $m>1+\frac{n-2}{2n}$.

math.AP↗
arXiv2018

Global existence and boundedness of weak solutions to a chemotaxis-stokes system with rotational flux term

In this paper, the three-dimensional chemotaxis-stokes system \begin{eqnarray*} \left\{\begin{array}{lll} \medskip n_{t}+u\cdot\nabla n=Δn^m-\nabla\cdot(n S(x,n,c)\cdot\nabla c),&x\inΩ,\ \ t>0, \medskip c_t+u\cdot\nabla c=Δc-nf(c),&x\inΩ,\ \ t>0, \medskip u_t+\nabla P=Δu +n\nablaϕ,&x\inΩ,\ \ t>0, \nabla\cdot u=0, &x\inΩ,\ \ t>0,, \end{array}\right. \end{eqnarray*} posed in a bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary is considered under the no-flux boundary condition for $n$, $c$ and the Dirichlect boundary condition for $u$ under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity $S(x,n,c)$ satisfies $S(x,n,c) 2$ for some non-decreasing function $\widetilde{S}\in C^{2}((0,\infty))$. In present work, it is shown that the weak solution is global in time and bounded while $m>m^\star(l)$, where \begin{eqnarray*} m^\star(l)= \left\{\begin{array}{lll} \medskip l-\frac{5}{6},\ &\mathrm{if}\ \frac{31}{12}\geq l>2, \medskip \frac{7}{5}l-\frac{28}{15},\ &\mathrm{if}\ l>\frac{31}{12}. \end{array}\right. \end{eqnarray*}

math.AP↗
arXiv2018

Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion

This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion \begin{eqnarray} \left\{\begin{array}{lll} n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(nχ(c)\nabla c),& x\inΩ,\ t>0, c_t+u\cdot\nabla c=Δc-nf(c),& x\inΩ,\ t>0, u_t+(u\cdot\nabla) u=Δu+\nabla P+n\nablaΦ,& x\inΩ,\ t>0, \nabla\cdot u=0,& x\inΩ,\ t>0 \end{array}\right. \end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $Ω\subset \mathbb{R}^3$ with smooth boundary. Here, $Φ\in W^{1,\infty}(Ω)$, $0<χ\in C^2([0,\infty))$ and $0\leq f\in C^1([0,\infty))$ with $f(0)=0$. It is proved that if $p>\frac{32}{15}$ and under appropriate structural assumptions on $f$ and $χ$, for all sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution.

math.AP↗
arXiv2017

John-Nirenberg Radius and Collapse in Conformal Geometry

Given a positive function $u\in W^{1,n}$, we define its John-Nirenberg radius at point $x$ to be the supreme of the radius such that $\int_{B_t}|\nabla\log u|^n<ε_0^n$ when $n>2$, and $\int_{B_t}|\nabla u|^2<ε_0^2$ when $n=2$. We will show that for a collapsing sequence in a fixed conformal class under some curvature conditions, the radius is bounded below by a positive constant. As applications, we will study the convergence of a conformal metric sequence on a $4$-manifold with bounded $\|K\|_{W^{1,2}}$, and prove a generalized Hélein's Convergence Theorem.

math.DG↗
arXiv2017

Conformal metric sequences with integral-bounded scalar curvature

Let $(M; g)$ be a smooth compact Riemiannian manifold without boundary and $g_{k}$ be a metric conformal to $g$. Suppose $vol(M; g_{k})+||R_{k}||_{L^{p}(M;g_{k})} < C$, where $R_{k}$ is the scalar curvature and $p > \frac{n}{2}$. We will use the 3-circle theorem and the John-Nirenberg inequality to study the bubble tree convergence of $g_{k}$.

math.DG↗
arXiv2017

Learning Spatiotemporal-Aware Representation for POI Recommendation

The wide spread of location-based social networks brings about a huge volume of user check-in data, which facilitates the recommendation of points of interest (POIs). Recent advances on distributed representation shed light on learning low dimensional dense vectors to alleviate the data sparsity problem. Current studies on representation learning for POI recommendation embed both users and POIs in a common latent space, and users' preference is inferred based on the distance/similarity between a user and a POI. Such an approach is not in accordance with the semantics of users and POIs as they are inherently different objects. In this paper, we present a novel spatiotemporal aware (STA) representation, which models the spatial and temporal information as \emph{a relationship connecting users and POIs}. Our model generalizes the recent advances in knowledge graph embedding. The basic idea is that the embedding of a $<$time, location$>$ pair corresponds to a translation from embeddings of users to POIs. Since the POI embedding should be close to the user embedding plus the relationship vector, the recommendation can be performed by selecting the top-\emph{k} POIs similar to the translated POI, which are all of the same type of objects. We conduct extensive experiments on two real-world datasets. The results demonstrate that our STA model achieves the state-of-the-art performance in terms of high recommendation accuracy, robustness to data sparsity and effectiveness in handling cold start problem.

cs.IR↗
arXiv2017

Asymptotics of Willmore minimizers with prescribed small isoperimetric ratio

We consider surfaces in ${\mathbb R}^3$ of type ${\mathbb S}^2$ which minimize the Willmore functional with prescribed isoperimetric ratio. The existence of smooth minimizers was proved by Schygulla (Archive Rational Mechanics and Analysis, 2012). In the singular limit when the isoperimetric ratio converges to zero, he showed convergence to a double round sphere in the sense of varifolds. Here we give a full blowup analysis of this limit, showing that the two spheres are connected by a catenoidal neck. Besides its geometric interest, the problem was studied as a simplified model in the theory of cell membranes, see e.g. Berndl, Lipowsky, Seifert (Physical Review A, 1991).

math.DG↗
arXiv2015

Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the $α_k$-harmonic map sequences with bounded uniformly $α_k$-energy, denoted by $\{u_{α_k}: α_k>1 \quad \mbox{and} \quad α_k\searrow 1\}$, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold is of a positive lower bound and the indices of the $α_k$-harmonic map sequence with respect to the corresponding $α_k$-energy are bounded, then, we can conclude that, if the blow-up phenomena occurs in the convergence of $\{u_{α_k}\}$ as $α_k\searrow 1$, the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence $u_k:(Σ,h_k)\rightarrow N$, where the conformal class defined by $h_k$ diverges, we also prove some similar results.

math.DG↗
arXiv2015

Extendability of conformal structures on punctured surfaces

For a smooth immersion $f$ from the punctured disk $D\backslash\{0\}$ into $\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence $r_k\to 0$, we show that the Riemannian surface $(D_r\backslash\{0\},g)$ where $g$ is the induced metric is conformally equivalent to the unit Euclidean punctured disk, for any $r\in(0,1)$. For a locally $W^{2,2}$ Lipschitz immersion $f$ from the punctured disk $D_2\backslash\{0\}$ into $\mathbb{R}^n$, if $\|\nabla f\|_{L^\infty}$ is finite and the second fundamental form of $f$ is in $L^2$, we show that there exists a homeomorphism $ϕ:D\to D$ such that $f\circϕ$ is a branched $W^{2,2}$-conformal immersion from the Euclidean unit disk $D$ into $\mathbb{R}^n$.

math.DG↗
arXiv2015

Some remarks on Willmore surfaces embedded in $\mathbb{R}^3$

Let $f:\mathbb{C}\rightarrow \mathbb{R}^3$ be complete Willmore immersion with $\int_Σ|A_f|^2<+\infty$. We will show that if $f$ is the limit of an embedded surface sequence, then $f$ is a plane. As an application, we prove that if $Σ_k$ is a sequence of closed Willmore surface embedded in $\mathbb{R}^3$ with $W(Σ_k)<C$, and if the conformal class of $Σ_k$ converges in the moduli space, then we can find a Möbius transformation $σ_k$, such that a subsequence of $σ_k(Σ_k)$ converges smoothly.

math.DG↗
arXiv2015

Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source

This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\inΩ,\ t>0$$ $$v_t=Δv-v+u,\quad x\inΩ,\ t>0$$ in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\geq1)$, where the functions $D(u)$ and $S(u)$ are supposed to be smooth satisfying $D(u)\geq Mu^{-α}$ and $S(u)\leq Mu^β$ with $M>0$, $α\in\mathbb{R}$ and $β\in\mathbb{R}$ for all $u\geq1$, and the logistic source $f(u)$ is smooth fulfilling $f(0)\geq0$ as well as $f(u)\leq a-μu^γ$ with $a\geq0$, $μ>0$ and $γ\geq1$ for all $u\geq0$. It is shown that if $α+2β<γ-1+\frac{2}{n}$, for $1\leqγ<2$ and $α+2β<γ-1+\frac{4}{n+2}$, for $γ\geq2$, then for sufficiently smooth initial data the problem possesses a unique global classical solution which is uniformly bounded.

math.AP↗
arXiv2015

Local Kondo entanglement and its breakdown in an effective two-impurity Kondo model

Competition between the Kondo effect and Ruderman-Kittel-Kasuya-Yosida interaction in the two-impurity Kondo problem can be phenomenologically described by the Rasul-Schlottmann spin model. We revisit this model from the quantum entanglement perspective by calculating both the inter-impurity entanglement and the local Kondo entanglement, the latter being the entanglement between a local magnetic impurity and its spatially nearby conduction electron. A groundstate phase diagram is derived and a discontinuous breakdown of the local Kondo entanglement is found at the singular point, associated concomitantly with a jump in the inter-impurity entanglement. An entanglement monogamy holds in the whole phase diagram. Our results identify the important role of the frustrated cross-coupling and demonstrate the local characteristic of the quantum phase transition in the two-impurity Kondo problem. The implications of these results for Kondo lattices and quantum information processing are also briefly discussed.

cond-mat.str-el↗