Existence of solutions with interior transition layers touching the boundary

For the case of Dirichlet boundary conditions, the existence of a solution with interior transition layer touching the boundary is proved.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 80–92.Original Russian Text Copyright © 2005 by I. V. Nedelko.This revised version was published online in April 2005 with a corrected issue number.     

Large time behavior of solutions to parabolic equations with Neumann boundary conditions

In this paper we are interested in the large time behavior as t→+∞ of the viscosity solutions of parabolic equations with nonlinear Neumann type boundary conditions in connection with ergodic boundary problems which have been recently studied by Barles and the author in [G. Barles, F. Da Lio, On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linèaire 22 (5) (2005) 521-541].     

Mathematical problems of the theory of elasticity of chiral materials for Lipschitz domains

By the potential method, we investigate the Dirichlet and Neumann boundary value problems of the elasticity theory of hemitropic (chiral) materials in the case of Lipschitz domains. We study properties of the single‐ and double‐layer potentials and of certain, generated by them, boundary integral operators. These results are applied to reduce the boundary value problems to the equivalent first and the second kind integral equations and the uniqueness and existence theorems are proved in various function spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct central impact of different elastic bodies of rotation

A direct central collision of two bodies of revolution is studied. A nonstationary mixed boundary-value problem with an unknown moving boundary is formulated. Its solution is represented by a series in term of Bessel functions. An infinite system of Volterra equations of the second kind for the unknown expansion coefficients is derived by satisfying the boundary conditions. The basic characteristics of the collision process are determined depending on the physic-mechanical properties of the bodies. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

We consider in this article the Cahn–Hilliard equation endowed with dynamic boundary conditions. By interpreting these boundary conditions as a parabolic equation on the boundary and by considering a regularized problem, we obtain, by the Leray–Schauder principle, the existence and uniqueness of solutions. We then construct a robust family of exponential attractors. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary value problems on manifolds with fibered boundary

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *‐algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A short proof of the twelve-point theorem

We present a short elementary proof of the following twelve-point theorem. Let M be a convex polygon with vertices at lattice points, containing a single lattice point in its interior. Denote by m (respectively, m*) the number of lattice points in the boundary of M (respectively, in the boundary of the dual polygon). Then m + m* = 12.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 117–120.Original Russian Text Copyright © 2005 by D. Repov, M. Skopenkov, M. Cencelj.This revised version was published online in April 2005 with a corrected issue number.     

Applying the dual operator formalism to derive the zeroth-order boundary function of the plasma-sheath equation

The article constructs the asymptotic solution of the Tonks-Langmuir integro-differential equation with an Emmert kernel, which describes the potential both in the bulk plasma and in a narrow boundary layer. Equations of this type are singularly perturbed, because the highest order (second) derivative is multiplied by a small coefficient. The asymptotic solution is obtained by the boundary function method. The second-order differential equation describing the behavior of the zeroth-order boundary function is investigated using the dual operator formalism — an analog of the conjugate operator in the linear theory. The application of this formalism has produced an asymptotic solution and has also made it possible to propose a number of homogeneous discrete three-point schemes for solving the equation. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 76–90, 2005.     

The destabilizing effect of boundary slip on Bénard convection

We investigate the influence of slip boundary conditions on the onset of Bénard convection in an infinite fluid layer. It is shown that the critical Rayleigh number is a decreasing function of the slip length, and therefore boundary slip is seen to have a destabilizing effect. Chebyshev‐tau and compound matrix formulations for solving the eigenvalue problem are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators

We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain H ⊂ ℝ _σ ^m . A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1131 – 1136, August, 2005.     

On the Turbulence Modelling of 3D-Atmospheric Boundary Layer Flow

The paper presents a mathematical and numerical investigation of the 3D–flow in the atmospheric boundary layer (ABL) over complex relief. The two–equation k - ε model is applied to account for the turbulence. The flow is also supposed to be viscous, incompressible and stationary. The boundary conditions are realized through the wall-functions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The solvability of the second initial boundary-value problem for the linear,time-dependent system of Navier-Stokes equations

In a class of functions with Holder-continuous derivatives unique solvability is is proved for the problem of determining a solution of the linear, time-dependent system of Navier-Stokes equations with boundary data , where are the direction cosines of the exterior normal to the boundary and are the components of the stress tensor.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 200–218, 1977.     

Boundary problems with meromorphic symbols in cylindrical domains

We show relative index formulas for boundary value problems in cylindrical domains and Sobolev spaces with different weights at ±∞. The amplitude functions are meromorphic in the axial covariable and take values in the space of boundary value problems on the cross section of the cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical Investigation of Wall Catalysis in High-Enthalpy Flows

A Navier-Stokes solver is applied to assess the effect of wall catalysis on the surface heat flux of hypersonic vehicles. The different catalytic boundary conditions are shortly discussed and results for a type IV interaction on a cylinder are presented and compared to experimental measurements. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact boundary controllability of unsteady flows in a network of open canals

By means of the general results on the exact boundary controllability for quasilinear hyperbolic systems, the author establishes the exact boundary controllability of unsteady flows in both a single open canal and a network of open canals with star configuration respectively. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pointwise estimates for Green's kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone

The paper deals with a mixed boundary value problem for the Stokes system in a polyhedral cone. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the sides of the polyhedron. The authors obtain regularity results for weak solutions in weighted L ₂ Sobolev spaces and prove point estimates of Green's matrix. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of thin polymer films

If we bond substrates like metal sheets by a thin polymer film we know about the existence of a boundary layer between the polymer and the substrate. The properties of this boundary layer differ from the bulk properties and influence the overall behavior of the bond. Here we present a mechanical theory based on a scalar-valued order parameter that allows us to describe these effects. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Boundary Conditions Modelling of the Fluid-Structure Interaction at Wind Tunnel Testing

The present paper discusses some newer developments, but concentrates mainly on the specification of the wall boundary conditions that are of importance for the calculation of the wind tunnel wall interference. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the asymptotic behavior of the principal eigenvalues of some elliptic problems

This paper is concerned with nonself-adjoint elliptic problems involving indefinite weights and boundary conditions of the Dirichlet, Neumann or Robin type. We study the asymptotic behavior of the principal eigenvalues, when the first order term (drift term) becomes larger and larger. The basic results of Berestycki et al. (Commun. Math. Phys., 253:451–480, 2005) are extended to the present context. Moreover, answers are provided to some open problems raised in Berestycki et al. (Commun. Math. Phys., 253:451–480, 2005).     

The Martin boundary and completion of Markov chains

Summary In this paper we treat a time-symmetrical Martin boundary theory for continuous parameter Markov chains. This is done by reversing the time sense of a Markov chainX _t in such a way as to obtain a dual Markov chain , and considering the two chains together. Various relations between the Martin exit boundaries and of these processes are studied. The exit boundary of , is in a sense an entrance boundary forX _t and vice versa. After a natural identification of certain points in and one can topologizeI in such a way thatboth X _t and have standard modifications in this space which are right continuous, have left limits, and are strongly Markov.Research supported in part at Stanford University, Stanford, California under AFOSR 0049.