DCDS-B
Convex spacelike hypersurfaces of constant curvature in de Sitter space
Joel Spruck Ling Xiao
We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(\kappa)=\sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity has at least one smooth solution (if $l=1$ or $l=2$ there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l=\sigma,\,1 \leq l < n$ in both de Sitter and Hyperbolic space.
keywords: de Sitter Hyperbolic space fully nonlinear. constant curvature asymptotic Plateau
DCDS
Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two
Xiaoli Li

This paper is devoted to the study of the initial-boundary value problem for density-dependent incompressible nematic liquid crystal flows with vacuum in a bounded smooth domain of $\mathbb{R}^2$. The system consists of the Navier-Stokes equations, describing the evolution of an incompressible viscous fluid, coupled with various kinematic transport equations for the molecular orientations. Assuming the initial data are sufficiently regular and satisfy a natural compatibility condition, the existence and uniqueness are established for the global strong solution if the initial data are small. We make use of a critical Sobolev inequality of logarithmic type to improve the regularity of the solution. Our result relaxes the assumption in all previous work that the initial density needs to be bounded away from zero.

keywords: Liquid crystals incompressible density-dependent vacuum global strong solution existence and uniqueness
JIMO
Second-order weak composed epiderivatives and applications to optimality conditions
Qilin Wang Xiao-Bing Li Guolin Yu
In this paper, one introduces the second-order weak composed contingent epiderivative of set-valued maps, and discusses some of its properties. Then, by virtue of the second-order weak composed contingent epiderivative, necessary optimality conditions and sufficient optimality conditions are obtained for set-valued optimization problems. As consequences, recent existing results are derived. Several examples are provided to show the main results obtained.
keywords: Set-valued optimization second-order optimality conditions. second-order weak composed contingent epiderivatives
JIMO
Some characterizations of robust optimal solutions for uncertain fractional optimization and applications
Xiang-Kai Sun Xian-Jun Long Hong-Yong Fu Xiao-Bing Li

In this paper, following the framework of robust optimization, we consider robust optimal solutions for a fractional optimization problem in the face of data uncertainty both in the objective and constraints. To this end, by using the properties of the subdifferential sum formulae, we first introduce some robust basic subdifferential constraint qualifications, and then obtain some completely characterizations of the robust optimal solutions of this uncertain fractional optimization problem. We show that our results encompass as special cases some optimization problems considered in the recent literature. Moreover, as applications, the proposed approach is applied to investigate weakly robust efficient solutions for multi-objective fractional optimization problems in the face of data uncertainty both in the objective and constraints.

keywords: Robust solutions subdifferential uncertain fractional optimization multi-objective optimization
DCDS-S
A periodic and diffusive predator-prey model with disease in the prey
Xiaoling Li Guangping Hu Zhaosheng Feng Dongliang Li
In this paper, we are concerned with a time periodic and diffusivepredator-prey model with disease transmission in the prey. Firstwe consider a $ SI $ model when the predator species is absent. Byintroducing the basic reproduction number for the $ SI $ model, weshow the sufficient conditions for the persistence and extinctionof the disease. When the presence of the predator is taken intoaccount, a number of sufficient conditions for the co-existence ofthe prey and predator species, the global extinction of predatorspecies and the global extinction of both the prey and predatorspecies are given.
keywords: Seasonality reaction-diffusion equations predator-prey model disease uniform persistence global extinction
DCDS-S
Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space
Baoquan Yuan Xiao Li
In this paper, we investigate the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations in three dimensions. We obtain that if $ \nabla_{h}u,\nabla_{h}\omega\in L^{1}(0,T;\dot{B}^{0}_{\infty,\infty})$ or $ \nabla_{h}u,\nabla_{h}\omega\in L^{\frac{8}{3}}(0,T;\dot{B}^{-1}_{\infty,\infty})$ then the solution $(u,\omega)$ can be extended smoothly beyond $t=T$.
keywords: Micropolar fluid equations blow-up criteria regularity criteria smooth solutions Besov space.
DCDS-B
Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching
Hongfu Yang Xiaoyue Li George Yin
This work is concerned with permanence and ergodicity of stochastic Gilpin-Ayala models involve continuous states as well as discrete events. A distinct feature is that the Gilpin-Ayala parameter and its corresponding perturbation parameter are allowed to be varying randomly in accordance with a random switching process. Necessary and sufficient conditions of the stochastic permanence and extinction are established, which are much weaker than the previous results. The existence of the unique stationary distribution is also established. Our approach treats much wider class of systems, uses much weaker conditions, and substantially generalizes previous results. It is shown that regime switching can suppress the impermanence. Furthermore, several examples and simulations are given to illustrate our main results.
keywords: positive recurrence Markov chain Stochastic permanence Gilpin-Ayala model stationary distribution.
BDIA
Increase statistical reliability without losing predictive power by merging classes and adding variables
Wenxue Huang Xiaofeng Li Yuanyi Pan

It is usually true that adding explanatory variables into a probability model increases association degree yet risks losing statistical reliability. In this article, we propose an approach to merge classes within the categorical explanatory variables before the addition so as to keep the statistical reliability while increase the predictive power step by step.

keywords: Association categorical data category merging statistical reliability predictive power
JIMO
Stability of solution mapping for parametric symmetric vector equilibrium problems
Xiao-Bing Li Xian-Jun Long Zhi Lin
This paper is concerned with the stability for a parametric symmetric vector equilibrium problem. A parametric gap function for the parametric symmetric vector equilibrium problem is introduced and investigated. By virtue of this function, we establish the sufficient and necessary conditions for the Hausdorff lower semicontinuity of solution mapping to a parametric symmetric vector equilibrium problem. The results presented in this paper generalize and improve the corresponding results in the recent literature.
keywords: parametric gap function. solution mapping semicontinuity Symmetric vector equilibrium problem continuity
DCDS-S
Well-posedness for the three-dimensional compressible liquid crystal flows
Xiaoli Li Boling Guo
This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.
keywords: existence and uniqueness. strong solution vacuum compressible Liquid crystals

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