On a bounded shear flow in a half-space

In this paper, a simple shear flow in a half-space, which has interesting properties from the point of view of boundary regularity, is described. It is a solution with a bounded velocity field to both the homogeneous Stokes system and the Navier–Stokes equation, and satisfies the homogeneous initial and boundary conditions. The gradient of the solution may become unbounded near the boundary. The example significantly simplifies an earlier construction by K. Kang, and shows that the boundary estimates obtained in a recent paper by the first author are sharp. Bibliography: 4 titles.     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

On the stability and the attraction domain of the stationary solution of a singularly perturbed parabolic equation with a multiple root of the degenerate equation

We construct and justify the asymptotics of a boundary layer solution of a boundary value problem for a singularly perturbed second-order ordinary differential equation for the case in which the degenerate (finite) equation has an identically double root. A specific feature of the asymptotics is the presence of a three-zone boundary layer. The solution of the boundary value problem is a stationary solution of the corresponding parabolic equation. We prove the asymptotic stability of this solution and find its attraction domain.     

Motion of a rigid body in a viscous fluid

We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.     

Boundary element solution of a scattering problem involving a generalized impedance boundary condition

A boundary element method is introduced to approximate the solution of a scattering problem for the Helmholtz equation with a generalized Fourier–Robin‐type boundary condition given by a second‐order elliptic differential operator. The formulation involves three unknown fields, but is free from any hypersingular integral. Existence and uniqueness of the solution are established using a Babuška inf–sup condition. When implementing the method, a lumping process allows to remove two fields from the formulation. The numerical solution has thus the same cost as the one of a problem relative to a usual Neumann boundary condition. Numerical tests confirm the ability of the method for solving this type of non‐standard boundary value problems. Copyright © 1999 John Wiley & Sons, Ltd.     

A bifurcation phenomenon in a singularly perturbed two-phase free boundary problem of phase transition

In this paper, we prove a bifurcation phenomenon in a two-phase, singularly perturbed, free boundary problem of phase transition. We show that the uniqueness of the solution for the two-phase problem breaks down as the boundary data decreases through a threshold value. For boundary values below the threshold, there are at least three solutions, namely, the harmonic solution which is treated as a trivial solution in the absence of a free boundary, a nontrivial minimizer of the functional under consideration, and a third solution of the mountain-pass type. We classify these solutions according to the stability through evolution. The evolution with initial data near a stable solution, such as the trivial harmonic solution or a minimizer of the functional, converges to the stable solution. On the other hand, the evolution deviates away from a non-minimal solution of the free boundary problem.     

Exact Analytic Solution of a Boundary Value Problem for the Falkner–Skan Equation

The Falkner–Skan equation, subject to appropriate physical boundary conditions arising from boundary layer theory, is exactly solved. The results obtained from this solution are compared with the numerical solution. The Blasius equation, subject to the same boundary conditions, is also solved exactly; the solution is compared with the earlier work on this equation. The analytic solution presented here agrees closely with the corresponding numerical results.     

Asymptotics of a Class of Singularly Perturbed Weak Nonlinear Boundary Value Problem with a Multiple Root of the Degenerate Equation

A singularly perturbed boundary value problem with weak nonlinearity in the case when the degenerate equation has a multiple root is studied. The asymptotic approximation of the solution is constructed by the modified boundary layer function method. Based on the comparison principle, there exist multizonal boundary layers in the neighborhood of the endpoints. The existence of a solution is proved by using the method of asymptotic differential inequalities.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

关于含小参数的拟线性椭圆型方程的狄立克雷问题

本文研究最高阶导数项含小参数的拟线性椭圆型方程的狄立克雷问题,在退化方程的特征是曲线和区域是凸域的一般情形下,给出构造一致有效渐近解的方法,并证明当小参数是充分小时,狄立克雷问题的解是存在和唯一.     

Smooth zero-contact-angle solutions to a thin-film equation around the steady state

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the “complete wetting” regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide “minimal” conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Approximate solution to a boundary value problem for the Lyapunov differential equation with a parameter

A special boundary value problem is studied for the Lyapunov differential equation which is used for investigation of the asymptotic properties of solutions to systems of periodic differential equations with a parameter. An algorithm is proposed for constructing an approximate solution to this boundary value problem, and conditions on the parameter are found under which the zero solution to the system is asymptotically stable.     

Neumann and Robin problems in a cracked domain with jump conditions on cracks

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series. Estimates for the absolute value of the solution of the boundary value problem and for the absolute value of the gradient of the solution are presented.     

Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end

We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form.     

Optimization of a nonlocal boundary control of vibrations of a string with a fixed end for an arbitrary time interval that is a multiple of 2l

We consider the optimization problem for a nonlocal boundary control of vibrations of an elastic string with fixed right end for an arbitrary sufficiently large time interval that is a multiple of twice the string length. We prove the existence and uniqueness of a generalized solution of the first boundary value problem and indicate an explicit expression for the solution. The optimal control is found in closed form.     

First mixed problem for a nonstrictly hyperbolic equation of the third order in a bounded domain

We study the classical solution of a boundary value problem for a nonstrictly parabolic equation of the third order in a rectangular domain of two independent variables. We pose Cauchy conditions on the lower base of the domain and the Dirichlet conditions on the lateral boundary. By the method of characteristics, we obtain a closed-form analytic expression for the solution of the problem. The uniqueness of the solution is proved.     

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ?Ω of the domain as the square root of the distance to ?Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

一类抛物方程初边值问题的精确解及其应用

本文建立了多层合采油藏压力分布新模型,并在无限大、有界定压和有界封闭三种边界条件下,求得了均质弱可压缩液体朝向具有井筒储集和表皮效应井渗滤时,各层内压力分布的精确表达式;给出了该压力解在油藏试井分析中的应用.