Global existence and decay estimates of the Boltzmann equation with frictional force: The general results

In this paper, we give the existence theory and the optimal time convergence rates of the solutions to the Boltzmann equation with frictional force near a global Maxwellian. We generalize our previous results on the same problem for hard sphere model into both hard potential and soft potential case. The main method used in this paper is the classic energy method combined with some new time–velocity weight functions to control the large velocity growth in the nonlinear term for the case of interactions with hard potentials and to deal with the singularity of the cross-section at zero relative velocity for the soft potential case.     

Spectra and ionization potential of cyanoacetylene

The spectrum of cyanoacetylene has been photographed in the vacuum ultraviolet region and its analysis made. The ionization potential has been calculated from an observed Rydberg series to be 11.6 eV. The first Rydberg band has been compared with the photoelectron spectrum of the molecule and conclusions about the initial orbital have been drawn.     

Open-surface MHD flow over a curved wall in the 3-D thin-shear-layer approximation

3-D thin-shear-layer equations for flows of conducting fluids in a magnetic field have been derived in orthogonal body-oriented coordinates and then applied to the analysis of MHD open-surface flows over a curved wall. Unlike the classic boundary-layer-type equations, present ones permit information to be propagated upstream through the induced magnetic field. Another departure from the classic theory is that the normal momentum equation keeps the balance between the pressure gradient term, and those related to gravity, centrifugal forces, and Lorentz force. Thus, the normal pressure variations are allowed. The model describes basic 3-D effects due to the wall curvature and spatial variations of the applied magnetic field. As a particular case, equations for flows with rotational symmetry have been derived. Numerical calculations were performed for open-surface flows over a body of revolution under conditions relevant to a fusion reactor (Hartmann number is 8500). Some specific flow patterns, such as flow thickening and spiral-type flows, have been observed and discussed. A special attention has been paid to the analysis of the magnetic propulsion as a tool for the active flow control by applying an electric current. It has been shown that depending on the applied current, the axial pressure gradient can act as an adverse pressure gradient or propulsion force.     

The planar Dirichlet problem for the Stokes equations

The Dirichlet problem for the Stokes equations is studied in a planar domain. We construct a solution of this problem in form of appropriate potentials and determine the unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence we determine a solution of the Dirichlet problem for a compressible Stokes system and a solution of a boundary value problem on a domain with cracks. Copyright © 2011 John Wiley & Sons, Ltd.     

A family of potentials for elliptic equations with one singular coefficient and their applications

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in the form of an integral equation. By using some properties of the Gaussian hypergeometric function, we first prove limiting theorems and derive integral equations concerning the densities of the double- and simple-layer potentials. The obtained results are then applied in order to find an explicit solution of the Holmgren problem for the multidimensional singular elliptic equation in the half of the ball.     

Mechanoelectromagnetic interaction in isotropic dielectrics with regard for the local displacement of mass

On the basis of the earlier proposed model of electromagnetothermomechanics of polarizable bodies, which takes into account the process of local displacement of mass, key equations for the corresponding scalar and vector potentials are written. The generalization of the Lorentz gauge, at which the equations for calculating the scalar and vector potentials of an electromagnetic field become uncoupled, is proposed. We write a resolving system of equations for potentials in the dimensionless form and obtain a parameter of interrelation of the processes of local displacement of mass and deformation. With the use of this system, the propagation of a plane harmonic wave in an infinite isotropic medium is investigated. It is shown that the model describes the dispersion of a modified elastic wave in the high-frequency region. The obtained results agree with known data presented in the literature and obtained from relations of the gradient theory of piezoelectrics.     

The Burgers equation with a random force and a general model for directed polymers in random environments

Summary. The study of the Burgers equation with a random force leads via a Hopf-Cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential. The latter can be studied by means of a general model of directed polymers in random environments with two point random potentials. These models exhibit a Gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and Burgers equations. Received: 18 July 1995 / In revised form: 5 August 1995     

On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials

This paper is concerned with well-posedness of the Cahn–Hilliard equation subject to a class of new dynamic boundary conditions. The system was recently derived in Liu–Wu (Arch. Ration. Mech. Anal. 233 (2019), 167–247) via an energetic variational approach and it naturally fulfils three physical constraints such as mass conservation, energy dissipation and force balance. The target problem examined in this paper can be viewed as a transmission problem that consists of Cahn–Hilliard type equations both in the bulk and on the boundary. In our approach, we are able to deal with a general class of potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double-obstacle potential. Existence, uniqueness and continuous dependence of global weak solutions are established. The proof is based on a novel time-discretization scheme for the approximation of the continuous problem. Besides, a regularity result is shown with the aim of obtaining a strong solution to the system.     

损伤粘弹性力学的广义变分原理及应用

从粘弹性材料的Boltzmann迭加原理和带空洞材料的线弹性本构关系出发,提出了一种损伤粘弹性材料具有广义力场的本构模型．应用变积方法得到了以卷积形式表示的泛函,并建立了损伤粘弹性固体的广义变分原理和广义势能原理．把它们应用于带损伤的粘弹性Timoshenko梁,得到了Timoshenko梁的统一的运动微分方程、初始条件和边界条件． 这些广义变分原理为近似求解带损伤的粘弹性问题提供了一条途径．     

Radial symmetry and monotonicity for an integral equation

In this paper we study radial symmetry and monotonicity of positive solutions of an integral equation arising from some higher-order semilinear elliptic equations in the whole space Rn. Instead of the usual method of moving planes, we use a new Hardy-Littlewood-Sobolev (HLS) type inequality for the Bessel potentials to establish the radial symmetry and monotonicity results.     

Pointwise gradient estimates

We survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear, possibly degenerate, elliptic and parabolic equations.     

Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials

Our aim in this article is to study the long time behaviour of a family of singularly perturbed Cahn‐Hilliard equations with singular (and, in particular, logarithmic) potentials. In particular, we are able to construct a continuous family of exponential attractors (as the perturbation parameter goes to 0). Furthermore, using these exponential attractors, we are able to prove the existence of the finite dimensional global attractor which attracts the bounded sets of initial data for all the possible values of the spatial average of the order parameter, hence improving previous results which required strong restrictions on the size of the spatial domain and to work on spaces on which the average of the order parameter is prescribed. Finally, we are able, in one and two space dimensions, to separate the solutions from the singular values of the potential, which allows us to reduce the problem to one with a regular potential. Unfortunately, for the unperturbed problem in three space dimensions, we need additional assumptions on the potential, which prevents us from proving such a result for logarithmic potentials. Copyright © 2004 John Wiley & Sons, Ltd.     

A molecular based model for polymer viscoelasticity: Intra- and inter-molecular variability

We develop dynamic equations for rubber viscoelasticity based on a stick-slip continuum molecular-based model. The model developed is a continuum tube reptation model in which a chemically cross-linked (CC) system of molecules act as constraint box per unit volume for a physically constrained (PC) system of molecules. The CC-system carries along the PC-system during instantaneous step deformations. The subsequent relaxation of the PC-system is determined by the configuration of the CC-system, its own configuration and confirmation, and external force fields. Conversely, the deformation of the PC-system acts as an internal variable affecting the deformations of the constraining CC-system. We model the relationship between these processes to derive a model of viscoelasticity in rubber deformation. In developing a relaxation process for the PC-system, we start from the fact that the PC-system is composed of long molecular chains. The dynamics of these molecular chains are developed by modelling them as chains of beads connected by springs, which represent inter-molecular potentials. Various segments of the molecular chains relax at different rates. In addition, variability in relaxation times across molecular chains is permitted.     

夹杂与基体对界面层螺旋位错的干涉效应

研究圆形夹杂与基体对有限厚度界面层螺旋位错的干涉问题。结合复变函数的分区亚纯函数理论、施瓦兹对称原理与柯西型积分运算,发展了多连通域联结问题的一个有效分析方法,将3个区域应力函数的联结问题化归为界面层应力函数的函数方程,并求得了显式级数解。利用该结果,研究与讨论了界面层螺旋位错能与位错力。     

薛定谔型椭圆方程的Morrey估计

利用与分数次积分相关的Olsen型不等式,我们得到了薛定谔型椭圆方程的内部Morrey估计, 其中位势函数满足反H\"older条件.     

Dissipative mechanism of a semilinear higher order parabolic equation in RN

It is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems. In this paper we investigate the dissipative mechanism for some semilinear fourth-order parabolic equations in the spaces of Bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor. While for second-order reaction–diffusion equations the dissipativeness mechanism has already been satisfactorily understood (see Arrieta et al. (2004), doi:10.1142/S0218202504003234 [7]), for higher order problems in unbounded domains it has not yet been fully developed. As shown throughout the paper, one of the main differences from the case of reaction–diffusion equations stems from the lack of a maximum principle. Thus we have to rely here on suitable energy estimates for the solutions. As in the case of second-order reaction–diffusion equations, we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness. Thus, the dissipative mechanisms in second-order and fourth-order equations are similar, although the lack of a maximum principle makes the proofs more difficult and the results not as complete.Finally, we make essential use of the sharp results of Cholewa and Rodriguez-Bernal (2012), doi:10.1016/j.na.2011.08.022 [12], on linear fourth-order equations with a very large class of linear potentials.     

Experimental and numerical study of the relaxation behavior of facial soft tissue

Challenges in computational simulation of the mechanical behavior of soft tissues and organs for clinical applications are related to the reliability of the models with respect to the anatomy, the mechanical interactions between different tissues, and the non linear (time dependent) force deformation characteristics of soft biological materials. In this paper a 3D finite element model of the face and neck, which has applications in surgical devices optimization and surgery planning, is presented. Bones, muscles, skin, fat, and superficial muscoloaponeurotic system (SMAS) were reconstructed from magnetic resonance images and their shape, constraints and interactions have been modeled according to anatomical, plastic and reconstructive surgery literature. Non linear time dependent constitutive equations are implemented in the numerical model, based on the Rubin-Bodner model. For the present calculations a simplified hyperelastic formulation has been used. The corresponding model parameters were selected according to previous work with mechanical measurements ex vivo on facial soft tissue. For determination of model parameters, in particular the ones corresponding to the time dependent behavior, an instrument for measuring the relaxation behavior of the face tissue in vivo was developed. The experimental set-up is described and results are presented for tests performed on different locations of the face (jaw, mid-face, parotid regions) and neck. The measured “long term” reaction force of the facial soft tissue is compared to numerical results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A complete steady state model of solute and water transport in the kidney

The purpose of this paper is to incorporate a detailed model, along with an optimized set of parameters for the proximal tubule, into J. L. Stephenson's current central core model of the nephron. In this model a set of equations for the proximal tubule are combined with Stephenson's equations for the remaining four tubules and interstitium, to form a complete nonlinear system of 34 ordinary differential and algebraic equations governing fluid and solute flow in the kidney. These equations are then discretized by the Crank-Nicholson scheme to form an algebraic system of nonlinear equations for the unknown concentrations, flows, hydrostatic pressure, and potentials. The resulting system is solved via factored secant update with a finite-difference approximation to the Jacobian. Finally, numerical simulations performed on the model showed that the modeled behavior approximates, in a general way, the physiological mechanisms of solvent and solute flow in the kidney.     

Functions of a complex variable in problems in the theory of elasticity with mass forces

The general equations of the theory of elasticity are reduced to an inhomogeneous fourth-order equation assuming that there is a linear dependence of the third component of the displacement vector on the third coordinate and that a mass force potential exists. The solution of this equation is presented, in particular, using two complex Kolosov–Muskhelishvili potentials. A third complex potential is introduced in addition to these. Using the three complex potentials, expressions are obtained for the components of the displacement vector and the stress and strain tensors that take account of mass forces. The application of the three potentials is analysed in problems in the theory of elasticity, and analytical solutions of several plane strain problems are presented.     

Uncertainty principle and kinetic equations

A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation.