arXiv subjects

Xinru Cao

Publications and source records attributed to Xinru Cao.

14 recordsLinked to original sources

arXiv2026

The Pozhidaev and Cantarini--Kac Constructions of Simple $n$-Lie Algebras: Distinctions and Realizations

Let $n\geq 3$, let $H\subseteq\mathbb{C}^n$ be any additive subgroup spanning $\mathbb{C}^n$, and let $0\ne t\in H$. We study Pozhidaev's central simple $n$-Lie algebra $P(H,t)=\widetilde{\mathcal A}(H,t)/\mathbb{C}e_0$ without a finite generation or discreteness assumption on $H$. Its inner derivation algebra is the simple generalized divergence-free Lie algebra $\mathcal S(0,0,n;t,H)$. We prove that its space of inner-equivariant symmetric products vanishes and that every $1/n$-derivation is a scalar multiple of the identity. Using these invariants, we show that $P(H,t)$ is not isomorphic to any simple nonabelian $n$-Lie algebra defined on the underlying spaces of the $S$, $W$, or $SW$ constructions recorded by Cantarini and Kac. We also show that $P(H,t)$ is the quotient by the constants of the derived algebra of an explicit $S$-algebra on $\mathbb{C}[H]$. Finally, we realize Pozhidaev's second construction $E(H)$ over $\mathbb{C}$ as a $W$-algebra.

math.RA↗

arXiv2026

Poisson $n$-Lie algebras: constructions and the structure of solvable algebras

In this paper, we develop a construction of Poisson $n$-Lie algebras that generalizes the Jacobian $n$-Lie construction. Using the Grassmann--Plücker relations, we derive necessary and sufficient conditions under which the resulting bracket defines a Poisson $n$-Lie algebra. We also prove that suitable quotients of tensor products of Poisson algebras carry natural Poisson $n$-Lie structures. Conversely, we give a tensor-type procedure that associates a Poisson algebra to a given Poisson $n$-Lie algebra. The quotient and converse constructions thus provide two systematic methods for relating Poisson algebras to Poisson $n$-Lie algebras. We further establish analogues of Engel's theorem and Lie's theorem and characterize solvability and nilpotency of Poisson $n$-Lie algebras in terms of their underlying associative and $n$-Lie structures. We introduce hypo-nilpotent ideals and investigate maximal such ideals in finite-dimensional solvable Poisson $n$-Lie algebras. Finally, we prove that the generalized eigenspaces of multiplication operators are ideals.

math.RA↗

arXiv2026

Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems

We consider the Neumann initial boundary value problem associated to the chemotaxis system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \big((u+1)^{m-1} u_x - u(u+1)^m v_x\big)_x & \text{in $(0, 1) \times (0, \infty)$}, \\ v_t = v_{xx} - v + u, &\text{in $(0, 1) \times (0, \infty)$}, \end{cases} \end{align} where $m \in \mathbb R$ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio $u(u+1)^m/(u+1)^{m-1}$ grows like $u^{2/n}$ for large $u$ with $n = \dim((0, 1)) = 1$. Nonetheless, we show that there is no critical mass phenomenon if $m \le -1$; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all $m \in (-\infty, -1] \cup (0, \infty)$ and even for all $m \in \mathbb R$ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding $W^{1, n} \hookrightarrow L^\infty$, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.

math.AP↗

arXiv2025

Reducing Averaging Time in Dual-comb Spectroscopy via Phase-Patterned Higher-Repetition-Rate Pulses

Dual-comb spectroscopy (DCS) is a powerful Fourier-transform spectroscopic technique that provides high-speed, high-resolution, and broadband measurements without moving parts. However, the high peak power of mode-locked pulses limits the photodetector's dynamic range, resulting in a low signal-to-noise ratio (SNR) per acquisition. While coherent averaging can improve SNR, it sacrifices temporal resolution and demands stringent system stability. Here, we introduce a novel concept to enhance SNR by using phase-patterned higher-repetition-rate combs. We reinterpret the self-imaging process of comb spectrum from a new perspective on mode interference among sub-pulse trains As a proof-of-concept, we densified two 250-MHz frequency combs to 12.5-MHz mode spacings via phase modulation and performed DCS on an $\mathrm{H^{13}C^{14}N}$ gas cell, and compared the results with an emulated conventional 12.5-MHz DCS, demonstrating a 17-fold increase in mode amplitude. This concept is expected to be combined with ultra-high repetition rate combs, such as microcombs, and thereby deployed in practical applications that typically require spectral sampling spacings from hundreds of MHz to GHz range.

physics.optics↗

arXiv2025

Finite-time blow-up in fully parabolic quasilinear Keller-Segel systems with supercritical exponents

We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller--Segel model \begin{align}\tag{$\star$}\label{prob:star} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u(u+1)^{q-1}\nabla v) & \text{in $Ω\times (0, T)$}, \\ v_t = Δv - v + u & \text{in $Ω\times (0, T)$} \end{cases} \end{align} in a ball $Ω\subset \mathbb R^n$ with $n\geq 2$. Previous results show that unbounded solutions exist for all $m, q \in \mathbb R$ with $m-q<\frac{n-2}{n}$, which, however, are necessarily global in time if $q \leq 0$. It is expected that finite-time blow-up is possible whenever $q > 0$ but in the fully parabolic setting this has so far only been shown when $\max\{m, q\} \geq 1$. In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that \eqref{prob:star} admits solutions blowing up in finite time if \begin{align*} m-q<\frac{n-2}{n} \quad \text{and} \quad \begin{cases} q < 2m & \text{if } n = 2, \\ q < 2m - \frac23 \text{ or } m > \frac23 & \text{if } n = 3, \end{cases} \end{align*} that is, also for certain $m, q$ with $\max\{m, q\} < 1$. As a key new ingredient in our proof, we make use of (singular) pointwise upper estimates for $u$.

math.AP↗

arXiv2024

Superlinear transmission in an indirect signal production chemotaxis system

In this paper, the indirect signal production system with nonlinear transmission is considered \[ \left\{ \begin{array}{lll} & u_t = Δu-\nabla\cdot(u \nabla v), \\ \displaystyle & v_t =Δv-v+w,\\ \displaystyle & w_t =Δw-w+ f(u) \end{array} \right. \] in a bounded smooth domain $Ω\subset \mathbb{R}^n$ associated with homogenous Neumann boundary conditions, where $f\in C^1([0,\infty))$ satisfies $0\le f(s) \le s^α$ with $α>0$. It is known that the system possesses a global bounded solution if $0<α<\frac 4n$ when $n\ge 4$. In the case $n\le 3$ and if we consider superlinear transmission, no regularity of $w$ or $v$ can be derived directly. In this work, we show that if $0<α< \min\{\frac 4n,1+\frac 2n\}$, the solution is global and bounded via an approach based on the maximal Sobolev regularity.

math.AP↗

arXiv2018

An interpolation inequality and its application in Keller-Segel model

In this paper, we first prove an interpolation inequality of Ehrling-type, which is an improvement of a special case to the well known Gargliardo-Nirenberg inequality. Then we apply it to study the classical Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=Δu-\nabla\cdot(u \nabla v), \\[6pt] \displaystyle v_t=Δv-v+u, \end{array} \right. \end{equation} in a bounded domain $Ω\subset\mathbb{R}^N$ ($N\ge 2$) with smooth boundary. It is known that for any $δ>0$, if $\int_Ωu^{\frac N2+δ}(\cdot,t)$ is bounded, then the solution is global and bounded. Here we show that the same conclusion holds for a weaker assumption: the equi-integrability of $\{\int_Ωu^\frac N2(\cdot,t)|~t\in(0,T_{\max})\}$ can prevent blow up.

math.AP↗

arXiv2017

Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller--Segel-Stokes system with competitive kinetics

This paper deals with the two-species Keller--Segel-Stokes system with competitive kinetics $(n_1)_t + u\cdot\nabla n_1 =Δn_1 - χ_1\nabla\cdot(n_1\nabla c)+ μ_1n_1(1- n_1 - a_1n_2)$, $(n_2)_t + u\cdot\nabla n_2 =Δn_2 - χ_2\nabla\cdot(n_2\nabla c) + μ_2n_2(1- a_2n_1 - n_2), c_t + u\cdot\nabla c =Δc - c + αn_1 +βn_2$, $u_t= Δu + \nabla P+ (γn_1 + δn_2)\nablaϕ$, $ \nabla\cdot u = 0$ under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^3$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+αn_1+βn_2$ is replaced with $-(αn_1+βn_2)c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that $μ_1,μ_2$ are sufficiently large. Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^\infty$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for $μ_1,μ_2$.

math.AP↗

arXiv2017

Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics

This paper considers the two-species chemotaxis-Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a three-dimensional bounded domain with smooth boundary. Both chemotaxis-fluid systems and two-species chemotaxis systems with competitive terms are studied by many mathematicians. However, there has not been rich results on coupled two-species-fluid systems. Recently, global existence and asymptotic stability in this problem with convection term in the fluid equation of the above system were established in the 2-dimensional case. The purpose of this paper is to give results for global existence, boundedness and stabilization of solutions to this system in the 3-dimensional case.

math.AP↗

arXiv2016

Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} \displaystyle n_t=Δn-\nabla\cdot(nS(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in Ω\times (0,T),\\ c_t=Δc-nc-u\cdot\nabla c , &(x,t)\inΩ\times (0,T),\\ u_t=Δu-κ(u\cdot\nabla)u+\nabla P+n\nablaϕ, &(x,t)\inΩ\times (0,T),\\ \nabla\cdot u=0,&(x,t)\inΩ\times (0,T), \end{array} \right.(\star) \end{equation} is considered under the no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $Ω\subset\mathbb{R}^N$ ($N=2,3$), $κ=0,1$. We assume that $S(x,n,c)$ is a matrix-valued sensitivity under a mild assumption such that $|S(x,n,c)|<S_0(c_0)$ with some non-decreasing function $S_0\in C^2((0,\infty))$. It contrasts the related scalar sensitivity case that $(\star)$ does not possess the natural {\em gradient-like} functional structure. Associated estimates based on the natural functional seem no longer available. In the present work, a global classical solution is constructed under a smallness assumption on $\|c_0\|_{L^\infty(Ω)}$ and moreover we obtain boundedness and large time convergence for the solution, meaning that small initial concentration of chemical forces stabilization.

math.AP↗

arXiv2016

Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} n_t=Δn-\nabla\cdot(n S(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in Ω\times (0,T), \displaystyle c_t=Δc-nc-u\cdot\nabla c, &(x,t)\inΩ\times (0,T), \displaystyle u_t=Δu-(u\cdot\nabla )u+\nabla P+n\nablaΦ,\quad \nabla\cdot u=0, &(x,t)\inΩ\times (0,T), \displaystyle \nabla c\cdotν=(\nabla n-nS(x,n,c)\cdot\nabla c)\cdotν=0, \;\; u=0,&(x,t)\in \partialΩ\times (0,T), n(x,0)=n_{0}(x),\quad c(x,0)=c_{0}(x),\quad u(x,0)=u_0(x) & x\inΩ, \end{array} \right. \end{equation} where $S\in (C^2(\barΩ\times [0,\infty)^2))^{N\times N}$, is considered in a bounded domain $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, with smooth boundary. We show that it has global classical solutions if the initial data satisfy certain smallness conditions and give decay properties of these solutions.

math.AP↗

arXiv2016

Boundedness in a three-dimensional chemotaxis-haptotaxis model

This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber \left\{ \begin{array}{llc} u_t=Δu-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w), &(x,t)\in Ω\times (0,T),\\ v_t=Δv-v+u, &(x,t)\inΩ\times (0,T),\\ w_t=-vw,&(x,t)\in Ω\times (0,T) \end{array} \right.\quad\quad(\star) \end{equation} under Neumann boundary conditions. Here $Ω\subset\mathbb{R}^3$ is a bounded domain with smooth boundary and the parameters $ξ,χ,μ>0$. We prove that for nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$, if $χ/μ$ is sufficiently small, ($\star$) possesses a global classical solution which is bounded in $Ω\times(0,\infty)$. We underline that the result fully parallels the corresponding parabolic-elliptic-ODE system.

math.AP↗

arXiv2015

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

This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type $ u_t=\nabla\cdot(ϕ(u)\nabla u)-χ\nabla\cdot(u\nabla v)+g(u)$, $τv_t=Δv-v+u$ in $Ω\times (0,T)$, subject to nonnegative initial data and homogeneous Neumann boundary condition, where $Ω$ is smooth and bounded domain in $\mathbb{R}^n$, $n\ge 2$, $ϕ$ and $g$ are smooth and positive functions satisfying $ks^p\leϕ$ when $s\ge s_0>1$, $g(s) \le as - μs^2$ for $s>0$ with $g(0)\ge0$ and constants $a\ge 0$, $τ,χ,μ>0$. It was known that the model without the logistic source admits both bounded and unbounded solutions, identified via the critical exponent $\frac{2}{n}$. On the other hand, the model is just a critical case with the balance of logistic damping and aggregation effects, for which the property of solutions should be determined by the coefficients involved. In the present paper it is proved that there is $θ_0>0$ such that the problem admits global bounded classical solutions, regardless of the size of initial data and diffusion whenever $\fracχμ<θ_0$. This shows the substantial effect of the logistic source to the behavior of solutions.

math.AP↗

arXiv2014

Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

In this paper, the fully parabolic Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=Δu-\nabla\cdot(u\nabla v), &(x,t)\in Ω\times (0,T),\\ v_t=Δv-v+u, &(x,t)\inΩ\times (0,T),\\ \end{array} \right. \qquad \qquad (\star) \end{equation} is considered under Neumann boundary conditions in a bounded domain $Ω\subset\mathbb{R}^n$ with smooth boundary, where $n\ge 2$. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find $\varepsilon_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\|u_0\|_{L^{\frac{n}{2}}(Ω)}<\varepsilon_0$ and $\|\nabla v_0\|_{L^{n}(Ω)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and approaches the constant steady state $(m,m)$ with $m:=\frac{1}{|Ω|}\int_Ω u_0$. Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.

math.AP↗