Three-dimensional periodic boundary layers

The method of successive approximations used by Eichelbrenner and A?kovíc¹ for the study of unsteady three-dimensional boundary layer flow has been extended to analyse the periodic boundary layers in three dimensions. The analysis, which is valid for oscillations of small amplitude, shows some special features such as “steady streaming” flow in the first-order cross-flow similar to the one that has been predicted and observed by Schlichting for two-dimensional periodic boundary layers.     

The effect of ghost forces for a quasicontinuum method in three dimension

We study the efect of"ghost forces"for a quasicontinuum method in three dimension with a planar interface."Ghost forces"are the inconsistency of the quasicontinuum method across the interface between the atomistic region and the continuum region.Numerical results suggest that"ghost forces"may lead to a negilible error on the solution,while lead to a fnite size error on the gradient of the solution.The error has a layer-like profle,and the interfacial layer width is of O(ε).The error in certain component of the displacement gradient decays algebraically from O(1)to O(ε)away from the interface.A surrogate model is proposed and analyzed,which suggests the same scenario for the efect of"ghost forces".Our analysis is based on the explicit solution of the surrogate model.     

Time reversed absorbing conditions

The aim of this note is to introduce the time reversed absorbing conditions (TRAC) in time reversal methods. These new boundary conditions enable one to “recreate the past” without knowing the source which has emitted the signals that are back-propagated. This new method does not rely on any a priori knowledge of the physical properties of the inclusion. We prove an energy estimate for the resulting non-standard boundary value problem. Two applications to inverse problems are given.     

On the core: complement-reduced game and max-reduced game

This paper presents two characterizations of the core on the domain of all NTU games. One is based on consistency with respect to “complement-reduced game” and converse consistency with respect to “max-reduced game”. The other is based on consistency with respect to “max-reduced game” and weak converse consistency with respect to “complement-reduced game”. Besides, we introduce an alternative definition of individual rationality, we name conditional individual rationality, which is compatible with non-emptiness. We discuss axiomatic characterizations involving conditional individual rationality for the core.     

A method of asymptotic constructions with improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation

The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter ε that takes any values from the half-open interval (0, 1]. For this type of linear problems, the order of the ε-uniform convergence (with respect to x and t) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge ε-uniformly at the rate of O(N ^?2ln² N + N ^?2 ₀), where N + 1 and N ₀ + 1 are the numbers of the mesh points with respect to x and t, respectively. On the x axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions.” Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width O(ε ln N)) the first derivative with respect to x is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to x), the classical implicit difference schemes approximating the first derivative with respect to x by the central difference derivative are applied. To improve the accuracy in t, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter ε.     

Boundary values of holomorphic semigroups of unbounded operators and similarity of certain perturbations

We obtain sufficient conditions for a “holomorphic” semigroup of unbounded operators to possess a boundary group of bounded operators. The theorem is applied to generalize to unbounded operators results of Kantorovitz about the similarity of certain perturbations. Our theory includes a result of Fisher on the Riemann-Liouville semigroup in L^p(0, ∞) 1 < p < ∞. In this particular case we give also an alternative approach, where the boundary group is obtained as the limit of groups in the weak operator topology.     

Fast Thermoviscous Convection

This paper studies the asymptotic structure of convection in an infinite Prandtl number fluid with strongly temperature-dependent viscosity, in the limit where the dimensionless activation energy 1/ε is large, and the Rayleigh number R, defined (essentially) with the basal viscosity and the prescribed temperature drop, is also large. We find that the Nusselt number N is given by N~CεR^1/5, where C depends on the aspect ratio a. The relative error in this result is O(R^?1/10ε^?1/4, ε^1/2, R^?2/5ε^?2, R^?2/20ε^?1/24), so that we cannot hope to find accurate confirmation of this result at moderate Rayleigh numbers, though it should serve as a useful indicator of the relative importance of R and ε. For the above result to be valid, we require R ? 1/ε⁵ ?1. More important, however, is the asymptotic structure of the flow: there is a cold (hence rigid) lid with sloping base, beneath which a rapid, essentially isoviscous, convection takes place. This convection is driven by plumes at the sides, which generate vorticity due to thermal buoyancy, as in the constant viscosity case (Roberts, 1979). However, the slope of the lid base is sufficient to cause a large shear stress to be generated in the thermal boundary layer which joins the lid to the isoviscous region underneath (though a large velocity is not generated); consequently, the layer does not “see” the shear stress exerted by the interior flow (at leading order), and therefore the thermal boundary layer structure is totally self-determined: it even has a similarity structure (as a consequence). This fact makes it easy to analyse the problem, since the boundary layer uncouples from the rest of the flow. In addition, we find an alternative scaling (in which the lid base is “almost” flat), but it seems that the resulting boundary layer equations have no solution, though this is certainly open to debate: the results quoted above are not for this case. When a free slip boundary condition is applied at the top surface, one finds that there exists a thin “skin” at the top of the lid which is a stress boundary layer. The shear stress changes rapidly to zero, and there exists a huge longitudinal stress (compressive/tensile) in this skin. For earthlike parameters, this stress far exceeds the fracture strength of silicate rocks.     

Oscillating random walk with a moving boundary

A discrete-time Markov chain is defined on the real line as follows: When it is to the left (respectively, right) of the “boundary”, the chain performs a random walk jump with distributionU (respectively,V). The “boundary” is a point moving at a constant speed γ. We examine certain long-term properties and their dependence on γ. For example, if bothU andV drift away from the boundary, then the chain will eventually spend all of its time on one side of the boundary; we show that in the integer-valued case, the probability of ending up on the left side, viewed as a function of γ, is typically discontinuous at every rational number in a certain interval and continuous everywhere else. Another result is that ifU andV are integer-valued and drift toward the boundary, then when viewed from the moving boundary, the chain has a unique invariant distribution, which is absolutely continuous whenever γ is irrational.     

Dyadic sets S: a dichotomy for indecomposable S-matrices

Dyadic sets S (see [3]) give rise to S-matrices, which are important in the investigation of modules over finite-dimensional algebras. If S admits only finitely many isoclasses of indecomposable S-matrices, “most” of the indecomposables can be described by means of a reduction to a well known special case. We determine the “missing” indecomposables.     

Expansions in Series of Products of Eigenfunctions

A method is presented to establish expansions of analytic functions in series of m-fold products of special functions of Mathematical Physics. The idea is to “multiply” vector-valued solutions of first order differential systems in a suitable way and to construct the first order differential system which the “product” satisfies. Then an expansion theorem for the corresponding Floquet eigenvalue problem can be proved.     

Théorèmes d'existence pour un modèle membranaire « proprement invariantde plaque non linéairement élastique

We establish the “local” existence of an injective solution to the nonlinear, “properly invariant”, membrane plate model, stated in [1] and [2], successively for the clamped plate submitted to forces parallel to its plane and for the plate submitted to a boundary condition of place of “extended” state.     

On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit L _p -viscosity solutions, which are in C ^1+α for an ${\alpha \in (0, 1)}$ . The equations have a special structure that the “main” part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.     

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

A minimum principle for superharmonic functions subject to interface conditions

Let D be a bounded domain in R² with smooth boundary. Let B₁, …, B_m be non-intersecting smooth Jordan curves contained in D, and let D′ denote the complement of ∪_{i ? 1}^mB_i respect to D. Suppose that $\text{u ? C}{}^2\text{(D′) ∩ C(}\text{D}\text{?}$) and Δu ? 0 in D′ (where Δ is the Laplacian), while across each “interface” B_i, i = 1,…, m, there is “continuity of flux” (as suggested by the theory of heat conduction). It is proved here that the presence of the interfaces does not alter the conclusions of the classical minimum principle (for Δu ? 0 in D). The result is extended in several regards. Also it is applied to an elliptic free boundary problem and to the proof of uniqueness for steady-state heat conduction in a composite medium. Finally this minimum principle (which assumes “continuity of flux”) is compared with one due to Collatz and Werner which employs an alternative interface condition.     

Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials

Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.     

Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over an exponentially stretching surface with suction in the presence of internal heat generation/absorption

In this paper, the problem of unsteady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid in the presence of thermal radiation, internal heat generation or absorption, and magnetic field over an exponentially stretching surface subjected to suction with an exponential temperature distribution is discussed numerically. The governing boundary layer equations are reduced to a system of ordinary differential equations. New numerical method using Mathematica has been used to solve such system after obtaining the missed initial conditions. Comparison of obtained numerical results is made with previously published results in some special cases, and found to be in a good agreement.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q _y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.     

Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.