Regularity of Potential Functions of the Optimal Transportation Problem

The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampère type. In this paper we introduce the notion of a generalized solution, and prove the existence and uniqueness of generalized solutions of the problem. We also prove the solution is smooth under certain structural conditions on the cost function.     

Global Cauchy Problem for the 2-D Magnetohydrodynamic-α Models with Partial Viscous Terms

The global Cauchy problem for the 2-D magnetohydrodynamic-α models with partial viscous terms is studied. The vanishing limit on α is also considered in this paper.     

Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli et al. (J Eur Math Soc (JEMS) 12(5):1151–1179, 2010). A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be zero along the boundary.     

The ∞-Eigenvalue Problem

. The Euler‐Lagrange equation of the nonlinear Rayleigh quotient is where is the minimum value of the quotient. The limit as of these equations is found to be where the constant is the reciprocal of the maximum of the distance to the boundary of the domain Ω. (Accepted May 19, 1998)     

The Inclined Wooding Problem

A constant-thickness thermal boundary layer is formed by the presence of uniform suction into a constant temperature hot surface bounding a porous medium—the Wooding problem. When the hot surface lies below the porous medium, the system is susceptible to thermoconvective instability. The present paper is concerned with how the classical linear stability analysis is modified by inclining the heated surface. Analysis is confined to disturbances in the form of transverse rolls because the equivalent analysis for longitudinal rolls may be described analytically in terms of that for the horizontal layer. The linear stability analysis is made difficult by the absence of a second bounding surface, and the method of multiple shooting is needed in order to obviate the consequence of having a stiff system of disturbance equations. Therefore, the computational domain is split into 5 or 10 subdomains. It is found that all modes of instability travel up the heated surface unless the surface is horizontal. The system is found to be linearly stable for all inclinations above \(31.85473^\circ \), a value which is remarkably close to that for the inclined Darcy-Bénard problem.     

动态规划求解空间双臂机器人非完整运动最优控制问题

通过自适应动态规划研究自由漂浮空间双臂机器人运动的最优控制问题．针对空间双臂机器人的非完整性运动,采用自适应动态规划(Adaptive Dynamic Programming, ADP)方法求解其最优控制问题．根据多体动力学理论,推导了载体位置、姿态均无控制条件下,双臂空间机器人满足的系统动量守恒关系的非完整约束方程,并将其转化为控制系统的状态方程,从而将双臂空间机器人的非完整运动规划问题转化为对非线性系统的控制问题．文中根据自适应动态规划网络结构,利用神经网络来近似性能指标函数,进而用龙格库塔法求解状态变量．并给出了适合该类问题的一种效用函数具体表达式,保证了空间双臂机器人到达期望位置后不再继续运动．实现了对空间双臂机器人非完整运动规划的最优控制．数值仿真实验验证了ADP对求解空间双臂机器人非完整运动规划最优控制问题的有效性．     

Coupled Solvent and Heat Transport of a Mixture of Swelling Porous Particles and Fluids: Single Time-Scale Problem

A three-spatial scale, single time-scale model for both moisture and heat transport is developed for an unsaturated swelling porous media from first principles within a mixture theoretic framework. On the smallest (micro) scale, the system consists of macromolecules (clay particles, polymers, etc.) and a solvating liquid (vicinal fluid), each of which are viewed as individual phases or nonoverlapping continua occupying distinct regions of space and satisfying the classical field equations. These equations are homogenized forming overlaying continua on the intermediate (meso) scale via hybrid mixture theory (HMT). On the mesoscale the homogenized swelling particles consisting of the homogenized vicinal fluid and colloid are then mixed with two bulk phase fluids: the bulk solvent and its vapor. At this scale, there exists three nonoverlapping continua occupying distinct regions of space. On the largest (macro) scale the saturated homogenized particles, bulk liquid and vapor solvent, are again homogenized forming four overlaying continua: doubly homogenized vicinal fluid, doubly homogenized macromolecules, and singly homogenized bulk liquid and vapor phases. Two constitutive theories are developed, one at the mesoscale and the other at the macroscale. Both are developed via the Coleman and Noll method of exploiting the entropy inequality coupled with linearization about equilibrium. The macroscale constitutive theory does not rely upon the mesoscale theory as is common in other upscaling methods. The energy equation on either the mesoscale or macroscale generalizes de Vries classical theory of heat and moisture transport. The momentum balance allows for flow of fluid via volume fraction gradients, pressure gradients, external force fields, and temperature gradients.     

The Regularity of Optimal Irrigation Patterns

A branched structure is observable in draining and irrigation systems, in electric power supply systems, and in natural objects like blood vessels, the river basins or the trees. Recent approaches of these networks derive their branched structure from an energy functional whose essential feature is to favor wide routes. Given a flow s in a river, a road, a tube or a wire, the transportation cost per unit length is supposed in these models to be proportional to s ^α with 0 < α < 1. The aim of this paper is to prove the regularity of paths (rivers, branches,...) when the irrigated measure is the Lebesgue density on a smooth open set and the irrigating measure is a single source. In that case we prove that all branches of optimal irrigation trees satisfy an elliptic equation and that their curvature is a bounded measure. In consequence all branching points in the network have a tangent cone made of a finite number of segments, and all other points have a tangent. An explicit counterexample disproves these regularity properties for non-Lebesgue irrigated measures.     

Problem of Optimal Control of the Turbulent Boundary Layer on a Permeable Surface in Supersonic Gas Flow

The problem of constructing the law of distribution of the normal component of the velocity of blowing to the turbulent boundary layer at supersonic flow velocities which ensure the minimum convective heat flow transmitted from the boundary layer to the surface is considered. The power of the control system calculated with regard to Darcy’s law of flow through a porous medium acts as the isoperimetric condition. The problemis solved using the Dorodnitsyn generalized integral relations. The numerical experiments carried out in the case of flow past a sphere showed the effectiveness of the optimal blowing laws as compared with the uniform law, namely, the gain in the minimized functional reaches 31.82%.     

线夹杂和线夹杂的相互作用

以短纤维复合材料为工程背景，本文利用线夹杂的工程计算模型以及无限平面中单夹杂的基本解，导出了线夹杂和线夹杂相互作用的平面问题的奇异积分方程。给出了夹杂端点的应力强度因子和夹杂界面应力的表达式，并作了具体的数值计算。     

The Effect of varying correlation lengths on Anomalous Transport

Conventional concepts for transport in porous media assume that the heterogeneous distribution of hydraulic conductivities is the source for the contaminant temporal and spatial heavy tail. This tailing, known as anomalous or non-Fickian transport, can be captured by the β parameter in the continuous-time random walk framework. This study shows that with the increase in spatial correlation length between these heterogeneous distributions of hydraulic conductivities, the transport’s anomaly reduces; yet, the β value is unchanged, suggesting a topological component of the conductivity field, captured by the β. This finding is verified by an analysis of the solute transport, showing that the changing conductivity values have a moderate effect on the transport shape.

     

The Anti-Plane Shear Problem in Nonlinear Elasticity Revisited

A classical problem in the framework of nonlinear elasticity theory is the characterization of the materials that may sustain a pure state of anti-plane shear in the absence of body forces. This problem has been solved by Knowles and by Hill in the framework of isotropic and incompressible elasticity in the seventies. Here we provide a simpler and shorter proof of these classical results. Moreover, we extend these results to nonlinear elastodynamics and we provide some new special solutions.     

The Optimal Shape of Riblets in the Viscous Sublayer

Our aim is to find the optimal shape of periodically distributed microstructures on surfaces of swimming bodies in order to reduce their drag. The model describes the flow in the viscous sublayer of the boundary layer of a turbulent flow. The microscopic optimization problem is reduced applying homogenization. In the reduced so-called macroscopic optimization problem we minimize the Navier constant subject to the boundary layer equations which are solved in a very small part of the original domain. Under the assumptions that the microstructures can be represented as smooth functions the sensitivity can be determined analytically. The optimization problem is then solved by a sensitivity based method (steepest descent with optimal step size) and the state equations are solved in each iteration with an external software. Our reduced model is validated by comparing the results from the homogenized model with those obtained by simulating the whole rough channel. An improved shape is found and a drag reduction up to 10% can be shown.     

飞机机翼气动外形优化设计研究

本文首先对某飞机原机翼外形进行了详尽的气动分析计算,然后确定了设计思路和方案,探讨了后掠角变化对机翼气动性能的影响,研究选定了减小外翼后掠角的机翼新平面形状,采用先进的CFD软件优化机翼的气动设计,根据不同设计思想进行了多个机翼的外形优化,包括新的翼剖面和弯扭配置,最后将优化设计结果与原机翼进行了对比,对比结果表明,以Q5-M2T和Q5-N2T为代表的优化结果取得了十分理想的改进效果,优化机翼提高了气动性能,机翼升阻比提高了20%-30%,满足了飞机载弹量增大后性能仍可以全面提高的设计要求。     

求解Reissner板弯曲问题的一种新方法

运用摄动理论首次将Ｒｅｉｓｓｎｅｒ板理论分解为一系列Ｋｉｒｃｈｈｏｆ板理论的叠加。这样，Ｒｅｉｓｓｎｅｒ板问题的解答就可通过在具体边界条件下求解一系列Ｋｉｒｃｈｈｏｆ板问题来获得。算例表明，本文方法简单，只需摄动两次就能接近Ｒｅｉｓｓｎｅｒ板理论的精确解，适合在实际工程中推广     

The Reconstruction Problem for the Euler-Poisson System in Cosmology

The motion of a continuum of matter subject to gravitational interaction is classically described by the Euler-Poisson system. Prescribing the density of matter at initial and final times, we are able to obtain weak solutions for this equation by minimizing the action of the Lagrangian which is a convex functional. Through this variational formulation, the reconstruction problem becomes very similar to an optimal transportation problem. Then we see that such minimizing solutions are consistent with smooth solutions of the Euler-Poisson system and enjoy some special regularity properties.     

The Role of Probabilistic Approaches to Transport Theory in Heterogeneous Media

A physical picture of contaminant transport in highly heterogeneous porous media is presented. In any specific formation the associated governing transport equation is valid at any time and space scale. Furthermore, the advective and dispersive contributions are inextricably combined. The ensemble average of the basic transport equation is equivalent to a continuous time random walk (CTRW). The connection between the CTRW transport equation, in a limiting case and the familiar advection–dispersion equation (ADE) is derived. The CTRW theory is applied to the results of laboratory experiments, field observations, and simulations of random fracture networks. All of these results manifest dominant non-Gaussian features in the transport, over different scales, which are accounted for quantitatively by the theory. The key parameter controlling the entire shape of the contaminant plume evolution and breakthrough curves is advanced as a more useful characterization of the transport than the dispersion tensor, which is based on moments of the plume. The role of probabilistic approaches, such as CTRW, is appraised in the context of the interplay of spatial scales and levels of uncertainty. We then discuss a hybrid approach, which uses knowledge of non-stationary aspects of a field site on a larger spatial scale (trends) with a probabilistic treatment of unresolved structure on a smaller scale (residues).