Primal-Dual Interior-Point Method for an Optimization Problem Related to the Modeling of Atmospheric Organic Aerosols

A mathematical model for the computation of the phase equilibrium related to atmospheric organic aerosols is presented. The phase equilibrium is given by the global minimum of the Gibbs free energy for a system that involves water and organic components. This minimization problem is equivalent to the determination of the convex hull of the corresponding molar Gibbs free energy function. A geometrical notion of phase simplex related to the convex hull is introduced to characterize mathematically the phases at equilibrium. A primal-dual interior-point algorithm for the efficient solution of the phase equilibrium problem is presented. A novel initialization of the algorithm, based on the properties of the phase simplex, is proposed to ensure the convergence to a global minimum of the Gibbs free energy. For a finite termination of the interior-point method, an active phase identification procedure is incorporated. Numerical results show the robustness and efficiency of the approach for the prediction of liquid-liquid equilibrium in multicomponent mixtures.Communicated by R. GlowinskiThis work was supported by US Environmental Protection Grant X-83234201. The second author was partially supported by Swiss National Science Foundation Grant PBEL2-103152.     

Fermionic entropy in BTZ space time

The thermodynamics of Dirac field is discussed in the backgrounds of 3 dimensional Banados-Teitelboim-Zanelli space time. The Dirac equation is solved under “quasi-periodic” boundary condition and the exact solution is obtained, from which the corresponding free energy and Fermionic entropy are calculated Project supported by the National Natural Science Foundation of China (Grant No. 9873013).     

Single Droplet Pattern in the Cylindrical Phase of Diblock Copolymer Morphology

The Ohta–Kawasaki density functional theory of diblock copolymers gives rise to a nonlocal free boundary problem. Under a proper condition between the block composition fraction and the nonlocal interaction parameter, a pattern of a single droplet is proved to exist in a general planar domain. A smaller parameter range is identified where the droplet solution is stable. The droplet is a set that is close to a round disc. The boundary of the droplet satisfies an equation that involves the curvature of the boundary and a quantity that depends nonlocally on the whole pattern. The location of the droplet is determined by the regular part of a Green’s function of the domain. This droplet pattern describes one cylinder in space in the cylindrical phase of diblock copolymer morphology.     

Fourier transform of the percolation free energy

Summary In this paper we compute the Fourier transform of the free energy of the percolation process. We apply the Fourier transform technique to rederive a result of Aizenman-Kesten-Newman that the derivative of the free energy is continuous.This research was supported by the Air Force Office of Scientific Research, Grant No. F49620 85 C0144Part of this work was also carried out at The University of Mississippi, Department of Mathematics, University, MS 38677, USA     

Global Existence and Exponential Stability for a Quasilinear Wave Equation with Memory Damping at the Boundary

In this paper, we focus on the global well-posedness of a quasilinear wave equation with a memory boundary condition. Under conditions on the geometry of the domain and the relaxation function describing the memory properties of the boundary, we obtain the existence, regularity and uniqueness of the global solution to the system. We prove also that the energy of the global solution to the system decays exponentially. The research was supported by the National Natural Science Foundation of China under Grant 60504001, Programa Nacional de Ayudas Para la Movilidad under Grant SB2003-0271 in Spain and the SIMUMAT projet of the CAM (Spain). The author is grateful to Prof. Enrique Zuazua for fruitful discussion.     

Droplet minimizers for the Cahn-Hilliard free energy functional

We prove theorems characterizing the minimizers for the Cahn-Hilliard free energy functional, which is used to describe the liquid vapor phase transition (or the 2 state magnetization transition). In particular, we exactly determine the critical density for droplet formation, and the geometry of the droplets.     

Schreier rewriting beyond the classical setting

Using actions of free monoids and free associative algebras, we establish some Schreiertype formulas involving ranks of actions and ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras. This work was supported by Natural Sciences and Engineering Research Council of Canada (Grant No. 227060-04), Yuri Bahturin, National Science Foundation (Grant No. DMS-0700811) and Russian Fund for Basic Research (Grant No. 08-01-00573), Alexander Olshanskii     

Asymptotics for a Symmetric Equation in Price Formation

We study the existence and asymptotics for large time of the solutions to a one dimensional evolution equation with non-standard right-hand side. The right-hand side involves the derivative of the solution computed at a given point. Existence is proven through a fixed point argument. When the problem is considered in a bounded interval, it is shown that the solution decays exponentially to the stationary state. This problem is a particular case of a mean-field free boundary model proposed by Lasry and Lions on price formation and dynamic equilibria. Maria P. Gualdani is supported by the NSF Grant DMS-0807636.     

Singularities on Free Surfaces of Fluid Flows

Isolated singularities on free surfaces of two-dimensional and axially symmetric three-dimensional steady potential flows with gravity are considered. A systematic study is presented, where known solutions are recovered and new ones found. In two dimensions, the singularities found include those described by the Stokes solution with a 120° angle, Craya's flow with a cusp on the free surface, Gurevich's flow with a free surface meeting a rigid plane at 120° angle, and Dagan and Tulin's flow with a horizontal free surface meeting a rigid wall at an angle less than 120°. In three dimensions, the singularities found include those in Garabedian's axially symmetric flow about a conical surface with an approximately 130° angle, flows with axially symmetric cusps, and flows with a horizontal free surface and conical stream surfaces. The Stokes, Gurevich, and Garabedian flows are exact solutions. These are used to generate local solutions, including perturbations of the Stokes solution by Grant and Longuet-Higgins and Fox, perturbations of Gurevich's flow by Vanden-Broeck and Tuck, asymmetric perturbations of Stokes flow and nonaxisymmetric perturbations of Garabedian's flow. A generalization of the Stokes solution to three fluids meeting at a point is also found.     

Energy identity of the heat flow of H -systems at finite singular time

It is well known that there exists a global solution to the heat flow of H-systems. If the solution satisfies a certain energy inequality, it is global regular with at most finitely many singularities. Under the same energy inequality, we can show the energy identity of the heat flow of H-systems at finite singular time. The most interesting thing in our proof is that we find the singular points can only occur in the interior of the set in some sense. This work was supported by the National Natural Science Foundation of China (Grant No. 10531020) and the Program of 985 Innovation Engineering on Information in Xiamen University (2004-2007) and New Century Excellent Talents of Xiamen Uiversity     

具有蒸发液滴两相介质近壁流动的数学分析

在双连续介质理论框架下，采用匹配渐进展开方法导出并求解了具有蒸发液滴的汽雾流中层流边界层方程，给出了控制汽雾流的相似判据。对于沿曲面的流动，边界层方程的形式取决于是否存在液滴的惯性沉积。给出了热钝体驻点附近蒸汽－液滴边界层的数值计算结果。它们表明：由于蒸发，在边界层内近壁处形成了一个无液滴区域；在该区上边界处，液滴半径趋于零而液滴数密度急剧增高。液滴蒸发及聚集的联合效应造成了表面热流的显著增加，甚至在自由来流中液滴质量浓度很低时此效应依然存在。     

Thin liquid droplet on top of a rotating non-isothermal liquid layer

Wafers are usually coated by using spin-coating, where centrifugal forces are used to spread a droplet on the rotating wafer. This flow is unstable to the fingering instability, where several segments of the wetting front spread faster than the average, resulting in several fingers. The liquid flows via the existing fingers while the area in between does not get coated. A precise experimental investigation is problematic, as the droplet has to be placed quite exactly in the center of the rotation. Replacing the wafer by a second liquid should lead to a parabolic-shaped free interface and the droplet should center itself due to gravity. Here, we derive a model for the free interfaces of a thin droplet on top of a rotating liquid by taking gravity, centrifugal forces, friction, (thermo-)capillary and line forces into account. Additionally, this setup is the simplest example of multiple coating, where several free interfaces and contact lines influence each other. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of aerosol spray characteristics for synthesis of sensor thin film from solution

A model is developed of aerosol spray for synthesis of sensor film from solution. The synthesis technique considered involves atomization of a solution of mixed salts in methanol, spraying of solution droplets, droplet deposition on a heated substrate, evaporation and chemical reaction to produce mixed oxides, and subsequent film growth. The precise control of oxide nanoparticle size distribution and inter-particle spacing in the film is crucial to achieving high sensitivity. These in turn largely depend on the droplet characteristics prior to impingement on the substrate. This paper focuses on the development of a model to describe the atomization and spray processes prior to the film growth. Specifically, a mathematical model is developed utilizing computational fluid dynamics solution of the equations governing the transport of atomized droplets from the nozzle to the substrate in order to predict droplet characteristics in flight. The predictions include spatial distribution of droplet size and concentration, and the effect on these characteristics of swirling inlet flow at the spray nozzle.     

Convergence behavior of interior-point algorithms

We show that most interior-point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition. These algorithms include all path-following algorithms and some potential reduction algorithms. The result also holds for the monotone complementarity problem if a strict complementarity solution exists. In general, the limit point is a solution that maximizes the number of its nonzero components among all solutions.Research supported in part by NSF Grant DDM-8922636, the Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.     

On thermodynamics and intrinsically equilibrated materials

Summary The existence and uniqueness of free energy functions is demonstrated for a class of materials broad enough to contain as special cases those of the theory of finite elasticity, the theory of hypo-elasticity, and the theory of internal state variables for which the path of evolution is invariant under rescalings of time. Dedicated toDario Graffi on his 70th Birthday Entrata in Redazione il 24 giugno 1975. This research was supported by the U. S. National Science Foundation under Grant GP-32951X2.     

Mesoscopic Analysis of Droplets in Lattice Systems with Long-Range Kac Potentials

We investigate the geometry of typical equilibrium configurations for a lattice gas in a finite macroscopic domain with attractive, long range Kac potentials. We focus on the case when the system is below the critical temperature and has a fixed number of occupied sites. We connect the properties of typical configurations to the analysis of the constrained minimizers of a mesoscopic non-local free energy functional, which we prove to be the large deviation functional for a density profile in the canonical Gibbs measure with prescribed global density. In the case in which the global density of occupied sites lies between the two equilibrium densities that one would have without a constraint on the particle number, a “droplet” of the high (low) density phase may or may not form in a background of the low (high) density phase. We determine the critical density for droplet formation, and the nature of the droplet, as a function of the temperature and the size of the system, by combining the present large deviation principle with the analysis of the mesoscopic functional given in Nonlinearity 22, 2919–2952 (2009).     

Feedback approximation of the minimum energy control for linear infinite-dimensional systems with zero terminal state

The paper studies the minimum energy control problem for linear infinite-dimensional systems with an unbounded input operator and zero terminal state. This problem is approximated by the minimum energy control problem with a small terminal state for which the solution is derived in feedback form. The operators which comprise the feedback are described in terms of differential relations which, depending on circumstances, involve Liapunov or Riccati differential equations. A detailed example illustrates how the general results apply to the wave equation with control in Dirichlet boundary condition.This work was supported by the Polish Ministry of National Education under Grant DNS-T/02/097/90-2.