具超音速边界可压Navier-Stokes方程解的指数衰减

考虑了二维空间上具超音速物理边界的可压Navier-Stokes方程的初边值问题.给定常数平衡态(ρ~*,0),得到了所考虑问题解的整体存在性.在平衡态附近的小扰动下,利用加权能量估计方法得到解的指数衰减性.     

带有对数非线性源的p-Kirchhoff方程解的整体存在性和衰减估计

考虑带有对数非线性源的p-Kirchhoff方程的初边值问题,此问题可用来描述热传播的过程和种群密度的演化.首先利用Galerkin方法,对数Sobolev不等式以及Gronwall不等式,再结合Lions引理,得到其局部解的存在性.同时引入修正泛函研究势井深度,结合势井理论,建立先验估计,得到解的整体存在性和衰减估计...     

一类带扩散的捕食模型非常数正解的存在性

本文考虑了一类带扩散的捕食模型的平衡态问题.首先给出了正解的先验估计,进而,分别借助于能量方法和拓扑度理论得出了因参数的变化而引起的非常数正解的不存在性和存在性结果.     

具有记忆项和非局部非线性项的板方程的整体吸引子

本文研究具有记忆项和非局部非线性项的板方程.首先利用近似的Faedo-Galerkin方法证得方程在初边值条件下解的适定性定理;其次通过先验估计并结合常用不等式证明该系统存在有界吸收集;最后利用Sobolev紧嵌入和收缩函数的方法证得解半群的渐近紧性,从而得到该系统整体吸引子的存在性.     

具有饱和与竞争项的捕食系统的全局分歧及稳定性

利用比较原理,分歧理论,特征值线性扰动理论,主要研究了一类具有饱和与竞争反应项的捕食-食饵系统在Dirichlet边界条件下的平衡态分歧解.首先给出了一个先验估计和局部分歧解存在的充分条件.然后对局部分歧解进行了全局延拓,得到了该系统平衡态的全局分歧解及其走向.最后讨论了局部分歧解的稳定性.     

带线性记忆和结构阻尼的非自治Kirchhoff型板方程的一致吸引子

本文试图研究带线性记忆和结构阻尼的非自治Kirchhoff型板方程的一致吸引子的存在性.为此,引入一个新的变量η,将先验估计与收缩函数的方法相结合,证明一致有界吸收集和一致渐近紧性,最终得到一致吸引子的存在性.     

一类具有非线性kirchhoff-sine-Gordon广义方程的整体吸引子的存在性

主要目的是利用Galerkin逼近法和先验估计来证明一类具有非线性阻尼和外源项的耗散型sine-Gordon-kirchhoff方程的整体吸引子的存在性,首先通过先验估计证明系统存在唯一的整体解,再证明系统存在有界吸收集和算子半群光滑性质,最后得到系统存在整体吸引子.     

一类带Michaelis-Menten收获项的Holling-Ⅳ型捕食-食饵模型的定性分析

本文对一类带Michaelis-Menten收获项的Holling-Ⅳ型捕食-食饵模型进行了定性分析.首先,利用极值原理和线性稳定性理论,得到了平衡态方程解的先验估计和正常数解的局部渐近稳定性;然后,借助分歧理论,给出了以d2为分歧参数,平衡态方程在正常数解U_1处的局部分歧,证明了在一定条件下,(d_2~j,U_1)处产生的局部分歧可以延拓成全局分歧.     

Boussinesq方程弱解的存在性和真空孤立现象（英文）

本文研究一类具有双阻尼非线性六阶Boussinesq方程的初边值问题,结合先验估计,利用Galerkin方法及紧性定理,得到该方程弱解的存在性.进一步,借助势井族的定义得到解的真空孤立现象,这是关于此方程的第一个结论.     

边界输入输出结构下非线性梁光滑解的存在性(英文)

本文研究一个描述梁振动的非线性模型,其非线性由物理条件(Hooke律)导致,主要研究该模型在边界输入输出结构下局部光滑解的存在性.首先应用发展方程理论证明相关线性系统存在光滑解,然后由一系列能量估计结合不动点定理证明所考察的非线性系统局部光滑解的存在性.     

Local well-posedness of the nonlinear Schrödinger equations on the sphere for data in modulation spaces

In this paper, the authors discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schrödinger on the sphere S². Exploring suitable a priori estimates, the authors prove the existence of solution for data whose regularity is s = 1/4.     

Sharp Threshold of Global Existence for a Nonlocal Nonlinear Schrödinger System in ℝ3

In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schr\"{o}dinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.     

Reflected Quadratic BSDEs Driven by G-Brownian Motions∗

In this paper, the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion (G-BSDE for short), with the generator growing quadratically in the second unknown. The authors obtain the existence by the penalty method, and some a priori estimates which imply the uniqueness, for solutions of the G-BSDE. Moreover, focusing their discussion at the Markovian setting, the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.     

Profile of bubbling solutions to a Liouville system

In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in R²${\mathbb{R}}^2$. Then we establish a uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated blowup point. The uniqueness result, as well as the local uniform estimates are crucial ingredients for obtaining a priori estimate, degree counting formulas and existence results for Liouville systems defined on Riemann surfaces.     

Global Existence, Uniqueness and Pathwise Property of Solutions to a Stochastic R¨ossler-Lorentz System

The authors integrate two well-known systems, the R¨ossler and Lorentz systems, to introduce a new chaotic system, called the Lorentz-R¨ossler system. Then, taking into account the effect of environmental noise, the authors incorporate white noise in both R¨ossler and Lorentz systems to have a corresponding stochastic system. By deriving the uniform a priori estimates for an approximate system and then taking them to the limit, the authors prove the global existence, uniqueness and the pathwise property of solutions to the Lorentz-R¨ossler system. Moreover, the authors carried out a number of numerical experiments, and the numerical results demonstrate their theoretic analysis and show some new qualitative properties of solutions which reveal that the Lorentz-R¨ossler system could be used to design more complex and more secure nonlinear hop-frequence time series.     

Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems

In this paper, some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions. The main technique is to use a standard finite element discretization on a coarse grid to approximate low frequencies and then to apply some linearized discretization on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local/parallel procedures. The theoretical tools for analyzing these methods are some local a priori and a posteriori error estimates for finite element solutions on general shape-regular grids that are also obtained in this paper.     

Strong solutions to the three-dimensional compressible viscoelastic fluids

The existence and uniqueness of the local strong solution to the three-dimensional compressible viscoelastic fluids near the equilibrium are established. In addition to the uniform estimates on the velocity, some essential uniform estimates on the density and the deformation gradient are also obtained.     

On the Cauchy problem for the magnetic Zakharov system

In this paper, we study the well-posedness results for the magnetic type Zakharov system. Such system describes the pondermotive force and magnetic field generation effects resulting from the nonlinear interaction between plasma-wave and particles. By using energy methods together with commutator estimate, we first derive a priori estimates for a regularized system. Then by approximation arguments, we obtain local existence results as well as global existence for small initial data.     

Local Solutions of Weakly Parabolic Quasilinear Differential Equations

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we lire able to prove L∞ - estimates.