Nonlinear Stability Problem of a Rotating Doubly Diffusive Porous Layer

Lyapunov direct method is applied to study the non-linear conditional stability problem of a rotating doubly diffusive convection in a sparsely packed porous layer. For a Darcy number greater than or equal to 1000, and for any Prandtl number, Taylor number, and solute Rayleigh number it is found that the non-linear stability bound coincides with linear instability bound. For a Darcy number less than 1000, for a Prandtl number greater than or equal to one, and for a certain range of Taylor number, a coincidence between the linear and nonlinear (energy) stability thermal Rayleigh number values is still maintained. However, it is noted that for a Darcy number less than 1000, as the value of the solute Rayleigh number or the Taylor number increases, the coincidence domain between the two theories decreases quickly.     

作旋转流动时非线性流体的改进模型

在圆环结构中研究拟塑性流体作圆形的Couette流动.流体的粘度依赖于对守恒方程有直接影响的剪切率,守恒方程采用谱方法求解.可以证明所采用的拟塑性模型,可以被适当地表示为典型的非线性流动.在早期研究中,为了方便数值计算,粘度表达式中只考虑了剪切率的二次项,与此不同,这里考虑了二次幂项.圆形Couette流动中弯曲的流线,造成离心的不稳定性,引起环形的漩涡,称之为Taylor漩涡.进而发现,随着拟塑性影响的增加,临界Taylor数下降.与已有圆形Couette流动的实验相比较,两者有着良好的一致性.     

Nonlinear Dynamics on the Plane and Integrable Hierarchies of Infinitesimal Deformations

A class of nonlinear problems on the plane, described by nonlinear inhomogeneous     -equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources), is described by Hamilton–Jacobi-type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are constructed by applying the quasiclassical     -dressing method.     

Hopf Bifurcation from Rotating Waves and Patterns in Physical Space

Summary. Hopf bifurcations from time periodic rotating waves to two frequency tori have been studied for a number of years by a variety of authors including Rand and Renardy. Rotating waves are solutions to partial differential equations where time evolution is the same as spatial rotation. Thus rotating waves can exist mathematically only in problems that have at least \bf SO (2) symmetry. In this paper we study the effect on this Hopf bifurcation when the problem has more than \bf SO (2) symmetry. These effects manifest themselves in physical space and not in phase space. We use as motivating examples the experiments of Gorman et al . on porous plug burner flames, of Swinney et al . on the Taylor-Couette system, and of a variety of people on meandering spiral waves in the Belousov-Zhabotinsky reaction. In our analysis we recover and complete Rand's classification of modulated wavy vortices in the Taylor-Couette system. It is both curious and intriguing that the spatial manifestations of the two frequency motions in each of these experiments is different, and it is these differences that we seek to explain. In particular, we give a mathematical explanation of the differences between the nonuniform rotation of cellular flames in Gorman's experiments and the meandering of spiral waves in the Belousov-Zhabotinsky reaction. Our approach is based on the center bundle construction of Krupa with compact group actions and its extension to noncompact group actions by Sandstede, Scheel, and Wulff. Received January 20, 1998; revised December 1, 1998     

一个具有非局部效应的非线性周期反应扩散方程的渐近形态

研究一个具有非线性-非局部反应的周期反应扩散系统.利用周期半流的渐近理论来讨论渐近波速c~*和周期行波解的存在性,证明参数c~*也是周期行波解的最小波速,并清晰描述解传播的阈值性质.最后给出渐近波速和最小波速c~*的估计.     

On Bounded Solutions of a Class of Nonlinear Integral Equations in the Plane and the Urysohn Equation in a Quadrant of the Plane

Ukrainian Mathematical Journal - We study a class of two-dimensional integral equations in the plane with monotonic nonlinearity. These equations have numerous applications in many various fields...     

经过修正的平面Couette流的非线性稳定性研究

本文讨论了经过修正的平面Couette流在二维扰动下的非线性稳定性性质,并同经过修正的平面Poiseuille流的非线性稳定性性质进行了比较.计算结果表明,对于有限振幅的扰动,平面Couette流比平面Poiseulle流更不稳定.     

Cutting Plane Algorithms for Nonlinear Semi-Definite Programming Problems with Applications

We will propose an outer-approximation (cutting plane) method for minimizing a function f X subject to semi-definite constraints on the variables XR ⁿ. A number of efficient algorithms have been proposed when the objective function is linear. However, there are very few practical algorithms when the objective function is nonlinear. An algorithm to be proposed here is a kind of outer-approximation(cutting plane) method, which has been successfully applied to several low rank global optimization problems including generalized convex multiplicative programming problems and generalized linear fractional programming problems, etc. We will show that this algorithm works well when f is convex and n is relatively small. Also, we will provide the proof of its convergence under various technical assumptions.     

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L²‐RH problems, we consider the construction of quarter plane solutions which are C¹ in time and C³ in space.     

Ornament Groups on a Minkowski Plane

We are engaged in classifying up to isomorphism of discrete subgroups of an affine transformation group on a plane (a two-dimensional space) preserving the Minkowski metric. It is proved that, for subgroups that do not coincide with Euclidean ones, the orbit of almost every point is everywhere dense.     

粘塑性流体在旋转圆盘上的流动

本文对两种情况导出了描述粘塑性流体在旋转圆盘上流动的基本方程.分别用摄动方法和数值方法得到了方程的解.这就有可能去计算薄膜的厚度分布.经计算发现有两种类型的厚度分布.对于粘度和屈服应力都与径向坐标r无关的粘塑性流体,厚度h随r的增加而减小.对于粘度和屈服应力都是时间和r的函数的Bingham流体,厚度h随r的增加而增加.     

非牛顿幂级数流体平面泊肃叶流的非线性稳定性*

非牛顿流体在现代工业生产中,广泛地作为加工的对象,因此非牛顿流体的流动稳定性问题有着重要的生产实际背景。本文选择在聚合物加工时将熔体压入成板口模时形成平面泊肃叶流这一生产背景,研究其非线性稳定性问题,计算其亚临界阈值。     

一类平面非线性自治系统分岔的计算及混沌

近年来非线住科学中混沌论的研究有了长足的进展，而微分动力系统的分岔问题常与温沌运动结有不解之缘。本文较详细地讨论了一类平面非线性自治系统的分岔问题，并指出由较简单分岔的相互作用最终会导致混沌运动的产生。     

平面Poiseuille流中弱非线性理论和二次失稳理论的关系*

文献[1]提出了平面Poiseuille流的二次失稳理论,本文则用弱非线性理论研究了同一问题.所得结果和二次失稳理论的结果是一致的,说明在平面Poiseuille流中弱非线性理论和二次失稳理论有内在联系.     

Locating a Central Hunter on the Plane

Protection, surveillance or other types of coverage services of mobile points call for different, asymmetric distance measures than the traditional Euclidean, rectangular or other norms used for fixed points. In this paper, the destinations are mobile points (prey) moving at fixed speeds and directions and the facility (hunter) can capture them using one of two possible strategies: either it is smart, predicting the prey’s movement in order to minimize the time needed to capture it, or it is dumb, following a pursuit curve, by moving at any moment in the direction of the prey. In either case, the hunter location in a plane is sought in order to minimize the maximum time of capture of any prey. An efficient solution algorithm is developed that uses the particular geometry that both versions of this problem possess. In the case of unpredictable movement of prey, a worst-case type solution is proposed, which reduces to the well-known weighted Euclidean minimax location problem. The work of the second and third authors was supported in part by a grant from Research Projects BFM2003-04062 and MTM2006-15054.