Layer Resolving Grids and Transformations for Singular Perturbation Problems [Reprint 2018 ed.] 9783110941944, 9783110460827

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Layer Resolving Grids and Transformations for Singular Perturbation Problems

LAYER RESOLVING GRIDS AND

TRANSFORMATIONS FOR

SINGULAR PERTURBATION PROBLEMS

VLADIMIR

D.

LISEIKIN

VSP BV

Tel: + 3 1 3 0 6 9 2 5 7 9 0

P.O. B o x 3 4 6

Fax: + 3 1 3 0 6 9 3 2 0 8 1 [email protected]

3700 AHZeist

www.vsppub.com

The Netherlands

© V S P B V 2001 First p u b l i s h e d in 2 0 0 1 ISBN 90-6764-346-7

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed

in The Netherlands

by Ridderprint

bv,

Ridderkerk.

CONTENTS Preface

ix

Chapter 1. Introduction to singularly perturbed problems 1.1. Introduction 1.2. Examples of singularly perturbed problems 1.2.1. Convection-diffusion problem 1.2.2. Momentum conservation laws 1.2.3. Prandtl equations 1.2.4. Problem of a thin beam 1.2.5. Problems of the shock wave structure 1.2.6. Burger's equation 1.2.7. One-dimensional steady reaction-diffusion-convection model 1.2.8. Orr-Sommerfeld problem 1.2.9. Diffusion-drift motion problem 1.3. Idealized problems 1.3.1. Semilinear problem 1.3.2. Weakly-coupled systems of ordinary differential equations 1.3.3. Autonomous equation 1.3.4. Equation with a power function multiplying the second derivative 1.3.5. General idealized problem 1.3.6. Invariants of equations 1.4. Singular functions 1.4.1. Definition of the singular functions 1.4.2. Examples of singular functions 1.4.3. Layer-type functions 1.5. Notion of layers 1.5.1. Definition of layers 1.5.2. Examples of layers 1.5.3. Partition of layers 1.5.4. Scale of a layer 1.5.5. Classification of layers 1.6. Basic approaches to analyze problems with a small parameter 1.6.1. Method of multivariable asymptotic expansions 1.6.2. Method of matched asymptotic expansions 1.6.3. Expansion via differential inequalities 1.6.4. Numerical methods 1.6.5. Method of layer-damping transformations 1.7. Comments

1 3 3 4 4 5 5 5 6 6 7 7 8 9 9 10 10 11 12 12 13 16 17 17 18 20 21 21 24 25 26 27 27 28 34

vi

contents

Chapter 2. Background for qualitative analysis 2.1. Introduction 2.2. Differential inequalities 2.2.1. Scalar problems 2.2.2. Systems of the second order 2.3. Theorems of inverse monotonicity 2.3.1. First order equations 2.3.2. Second order equations 2.4. Requirements imposed on estimates of the derivatives 2.4.1. Formulation of an optimal univariate transformation 2.4.2. Necessary bounds for the first derivative 2.4.3. Bounds on the higher derivatives 2.4.4. Uniform bounds on the total variation 2.5. Inequality relations 2.6. Comments

37 37 37 40 43 43 47 55 56 56 58 59 62 63

Chapter 3. Estimates of the solution derivatives to semilinear problems 3.1. Introduction 3.2. Initial problem 3.2.1. Smooth terms 3.2.2. Nonsmooth terms 3.3. Second order equations 3.3.1. Strong ellipticity 3.3.2. Problem with the condition f(x,u) = xg(x,u) 3.3.3. Problem of population dynamics theory 3.3.4. Generalization to mixed boundary conditions and dependence on e 3.4. Equation with a power function affecting the second derivative 3.4.1. Power singularities 3.4.2. Exponential singularity 3.5. Generalization to elliptic and parabolic equations 3.5.1. Estimates of the solution derivatives 3.6. Comments

65 65 65 70 72 72 97 103 105 106 106 110 111 112 113

Chapter 4. Problems for ordinary quasilinear equations 4.1. Introduction 4.2. Autonomous boundary value problem 4.2.1. Preliminary results 4.2.2. Boundary layers 4.2.3. Interior layers 4.3. Nonautonomous Equation 4.3.1. Estimates of the first derivative 4.3.2. Graphical chart for localizing the layers 4.3.3. Example of the problem 4.4. Analysis of the limit solution 4.4.1. Properties of the limit solution 4.5. Comments

115 115 115 117 119 127 127 142 143 144 145 148

contents

vii

Chapter 5. Systems of ordinary differential equations 5.1. Introduction 5.2. Equations of the first order 5.2.1. Initial-value problem without turning points 5.2.2. Equation with a turning point 5.3. Semilinear equations of the second order 5.3.1. Estimates of the derivatives 5.3.2. Estimates of the first derivative 5.3.3. Estimates of the higher derivatives 5.4. Derivative estimates and location of the shock layer 5.5. Comments

149 149 149 152 154 156 158 169 180 182

Chapter 6. Generation of transformations and layer-resolving grids 6.1. Introduction 6.2. Implicit generation of layer-damping transformations 6.2.1. One-dimensional case 6.2.2. Extension to multidimensions 6.3. Explicit generation of layer-damping transformations 6.3.1. Basic majorants 6.3.2. Generation of univariate local stretching transformations 6.3.3. Basic layer-damping transformations 6.3.4. Construction of basic intermediate transformations 6.4. Algebraic method for generating coordinate transformations 6.4.1. Lagrange interpolation 6.4.2. Hermite interpolation 6.4.3. Control of grid resolution 6.5. Comments

183 185 185 191 193 193 197 202 211 214 218 220 222 224

Chapter 7. Analysis of numerical algorithms 7.1. Introduction 7.2. Relations between the solution and truncation errors 7.2.1. Initial problem 7.2.2. Two-point boundary value problem 7.3. Uniform convergence for initial-value problems 7.3.1. Equation with a turning point 7.3.2. Equation without a turning point 7.4. Uniform convergence for two-point boundary value problems 7.4.1. Equation without a boundary turning point 7.4.2. Boundary turning points 7.4.3. Interior turning points 7.4.4. General case 7.4.5. Problem with the condition f(x, u) = xg(x, u) 7.4.6. Uniform convergence of the numerical friction 7.5. Problem with a variable coefficient before the second derivative 7.5.1 Case 0 < a ( 0 ) < 1 7.5.2. Case a (0) > 1 7.5.3. Case a (0) = 1

227 227 228 230 231 231 233 233 234 239 243 244 246 254 256 258 259 262

viii

contents

7.5.4. Algorithm for generating layer-resolving grids 7.5.5. Generalization to systems 7.5.6. Generalization to elliptic and parabolic equations 7.5.7. Uniform convergence at the points of a uniform grid 7.6. Comments

263 263 264 266 268

References

270

Index

283

Preface One of the most pressing problems faced by developers of numerical codes is that concerned with a creation of adequate computational techniques aimed at the numerical solution of problems with singularities. Typical examples of the problems are presented by singularly perturbed equations having a small parameter e affecting the higher derivatives. These problems arise frequently in many practical applications such as semiconductor theory, fluid dynamics, seismology, chemistry, geophysics, nonlinear mechanics, etc. A distinctive feature of the singularly perturbed equations is that their solutions and/or the solution derivatives have intrinsic narrow zones (boundary and interior layers) of large variations in which they jump from one stable state to another or to prescribed boundary values. Such situation occurs in many physical, chemical, biological, and sociological phenomena. In physics, for example, this happens in viscous gas flows in the zones near the boundary layers where the viscous flow jumps from the boundary values prescribed by the condition of adhesion to the inviscid flow or in the zones near the shock wave where the flow jumps from a subsonic to supersonic state. In chemical reactions the rapid transition from one state to another is typical for solution processes. In biology such sharp changes occur in population genetics. Typical examples of rapid transitions in sociology give revolutions and corresponding changes of political institutions. The singularly perturbed problems having intrinsic boundary and interior narrow layers where the derivatives of their solutions with respect to the coordinate orthogonal to the corresponding layer reach very large values when the parameter is small are not readily treated by standard analytical and numerical methods. As a consequence much research efforts are aimed at the development of better techniques adjusted to the solution of these problems. Presently there are four special basic approaches to treat the problems with boundary and interior layers. The classical one relies on expansions of solutions with a series of singular and slow changing functions. The second technique applies some specified difference approximations of equations. The third one is based on the implementation of layer-resolving or fitted grids. In the fourth approach nonuniform transformations are introduced as locally stretching coordinates. These transformations can also be used naturally for the generation of grids clustered in the vicinity of the layers. The major requirement imposed on the locally stretching

X

Preface

coordinates is t h a t they should be layer-damping, i.e. the singularities of solutions with respect to these coordinates should be eliminated or at least mitigated. The approach using the locally stretching coordinate transformations appears to be more effective in comparison with other techniques, because it requires a rather rough information about the necessary qualitative properties of solutions and, what is very important, it enables one to interpolate the numerical solutions e-uniformly to the entire physical region. As a result, the solutions interpolated from the grid nodes converge e-uniformly to the accurate ones on the whole physical domain including the layers. The approach of the layer-damping coordinate transformations to treat the singularly perturbed equations is rather young and it is still growing fast and new studies are continually being added to the field of the qualitative analysis of solutions, development of codes, and application to more important practical problems. Therefore there exists an evident need of students, researchers, and practitioners in applied mathematics and industry for new books which coherently complement the existing ones with the description of new developments of the approach and concomitant areas of its technology. The objective of the monograph is to give a clear, comprehensive, and easily learned description of the qualitative properties of solutions to singularly perturbed problems and of the essential elements, methods, and codes of the technology adjusted to the numerical solution of equations with singularities by an application of the layer-damping coordinate transformations and corresponding layer-resolving grids. In accordance with this goal the first part of the book is confined to the analytical study of estimates of the solutions and their derivatives in the layers of singularities and to compiling and manifesting suitable techniques to extend the results presented. The second part of the book describes a technique to build the coordinate transformations eliminating the boundary and interior layers. The third part reviews some numerical algorithms based on the technique developed for the generation of the layer-damping coordinate transformations and the corresponding layerresolving meshes. The book is largely devoted to a qualitative study of solutions to singularly perturbed problems and to a detailed review of those important aspects concerned with the development of the numerical techniques for equations with singularities which have not been sufficiently covered in the written monographs. Special attention is paid to the description, proof, and substantiation of estimates of the solution derivatives of nonlinear

Preface

xi

and the so-called bisingular problems which have additional singularities from the reduced problems. The bisingular problems are widespread in practical applications, for example, they are presented by the equations of the second order with turning points as in the case of the Navier-Stokes equations with the boundary condition of adhesion. As for the nonlinear problems, they dominate in practical considerations. These problems are the most important for applications and the most difficult for the pure analytical and numerical studies. However, for the method of the layer-damping transformations the prospect is more promising since it combines both the analytical and numerical approaches. Besides this, the current analytical tools allow one to obtain the necessary estimates of the solution derivatives quite efficiently. Therefore, the book may stimulate the efforts of researchers in establishing the qualitative properties of solutions to more complicated problems and in the development of the numerical codes which are required in real practical applications in contrast to the present numerical developments which are chiefly oriented on the numerical solution of problems with exponential layers only what largely corresponds to the condition t h a t the solutions of the reduced equations do not have singularities. The book includes a description of some elements of a multidimensional algorithm which can be used to generate suitable coordinate transformations and grids not only for singularly perturbed problems but for arbitrary problems with singularities. Imperative for this purpose is the application of interactive technologies. Though a somewhat brilliant prospect of the approach advocated in the book is evident for the author, however, there is a need in some critical mass of the knowledge of its tools, opportunities, and spectacular results and as well as in the availability of the information about the approach described completely and compactly in order the process of its development became attractive for other researchers. Perhaps this book may appear to be t h a t sufficient contribution by which the necessary critical mass is attained, thus arousing an interest of mathematicians in attacking singularly perturbed problems with this promising and advancing approach. The first chapter acquaints the reader with some specific singularly perturbed problems and introduces him to the difficulties concerned with obtaining solutions to such problems. Chapter 1 also discusses the notion of the invariants of singularly perturbed differential equations, which in reality are the basic elements controlling the qualitative behavior of their solutions and the location of their layers. Some laws connecting the invariants and the estimates of the derivatives are expounded in the chapter. Various types of functions with singularities and layers of different kinds

xii

Preface

are demonstrated and classified. The chapter also describes the most popular approaches aimed at overcoming the difficulties pertinent to the numerical solution of singularly perturbed equations. Chapter 2 provides the reader with a general background necessary for the qualitative analysis of singularly perturbed problems. The analysis is needed for the purpose of the construction of layer-damping coordinate transformations and it concludes in obtaining some solution derivative estimates. The chapter reviews the most important tools to get such estimates such as Nagumo's theorem, the method of barrier functions, and the theorems of inverse monotonicity. It ends with an estimate of the variation of the solution to a two-point singularly perturbed boundary value problem. Chapters 3-5 are devoted to obtaining estimates of the solution derivatives for various types of singularly perturbed problems. The estimates enable one to come up with a priori layer-damping coordinate transformation and corresponding layer-resolving mesh, whereas in general such transformations and meshes can be generated only adaptively. However, the application of interactive procedures with the basic intermediate transformations formed through the basic locally stretching functions allows one to generate the efficient layer-damping coordinate transformations and layer-resolving grids even without having any preliminary information about the qualitative features of the evolving solutions. Much attention in Chaps. 3-5 is paid to nonlinear and bisingular equations. The bisingular equations are especially difficult to provide the pure analytical and numerical analysis since they have additional singularities caused by the reduced equations, such as the equations of the second order whose coefficients before the first derivatives are not separated from zero. The qualitative analysis given in these chapters reveals three new boundary and interior layer functions which in contrast to the well-known exponential one are of power types. These very functions abound in nonlinear and bisingular problems. The theoretical results on the analytical solutions exhibited in Chaps. 3-5 are used in Chap. 6 to design the layer-damping coordinate transformations which can eliminate singularities of solutions to singularly perturbed problems and to construct e-uniform numerical algorithms. The layer-damping coordinate transformations are generated through the use of some procedures over four basic locally contracting mappings corresponding to the basic exponential and power singular functions. These functions are described in Chap. 1 as representatives of the general solutions of some model singular perturbation problems. Since the derivatives with respect to the new coordinates appear to be e-uniformly bounded,

Preface

xiii

the structured grids obtained by mapping uniform grids from a standard computational domain into the physical area by the layer-damping coordinate transformations are optimally distributed and provide the e-uniform convergence and interpolation of the numerical solutions to the accurate ones. Chapter 6 also reviews a grid generation approach based on a combination of the algebraic method and the construction of the intermediate coordinate transformations with the help of the standard layer-damping functions. This approach is suitable for the generation of both hexahedral and tetrahedral (quandrangular and triangular in two dimensions) grids. The technique reviewed in the chapter allows one to generate adequate grids for the numerical solution of multidimensional singularly perturbed and sigular equations. Theoretical substantiation of the efficiency of the layer-damping coordinate transformations and corresponding layer-resolving grids to obtain the numerical solutions to problems with singularities is given in Chap. 7. The mapping technology based on the introduction of the coordinate transformations allows one to formulate two approaches to the numerical solution of problems with singularities: 1. by approximating the equations on a nonuniform grid in the physical domain and 2. by approximating the transformed equations in the computational domain on a uniform grid. Both approaches are discussed in the chapter. For versatile problems considered and analyzed the ^-uniform convergence and interpolation of the numerical solutions is proved. The chapter also explains some anomalies related to the e-uniform convergence of the numerical solutions at the nodes of uniform coarse grids. The monograph ends with a list of bibliography. The author is greatly thankful to Galya Liseikin and Aleksey Liseikin who assisted him with preparing the figures and text of the manuscript in lATgX code.

Chapter 1

Introduction to Singularly Perturbed Problems 1.1.

Introduction

This chapter gives an introduction to the subject of singularly perturbed problems. It delineates some types of concrete problems, their areas of applications and the most popular approaches aimed at their solution. The chapter also presents some notions, classifications, and examples of singular and layer type functions and the layers of their singularities pertinent to the singular perturbation theory. There is a significant interest in theoretical and numerical studies of equations with singularities, in particular, singularly perturbed equations, i.e. ones with a small positive parameter e in a coefficient before the higher derivatives. These equations are widespread in practical applications. For example, such equations model viscous flows, where the small parameter is typically the reciprocal of the nondimensional Reynolds number Re, describe problems of elasticity, where the parameter represents the shell thickness, or simulate flows of liquid in regions having orifices with a small diameter. As a rule, the solutions of these problems have highly localized regions (boundary and interior layers) of rapid variation in the center of which the solution derivatives reach magnitudes of order £~k, k > 0, thus tending to infinity when e approaches zero. Outside the layers the derivatives are estimated by a constant M independent of the parameter e. Specifically, these features of the singularly perturbed equations cause serious difficulties for the efficient studies of their solutions by standard methods when the parameter e is very small. At present, there are analytical, numerical, and combined approaches applied to the analysis of singularly perturbed problems. The analytical approach relies on asymptotic expansions: multivariable, matched, and those ones performed by shooting techniques; while the numerical technology uses fitted approximations, layer-damping coordinate transformations, and dynamically adaptive and a priori generated grids. Currently, the pure analytical and numerical techniques have demonstrated the most spectacular results in applications to the problems whose solutions have singularities of exponential types only. However, such singularities are not typical in practical applications, moreover, they appear in a very narrow

2

1. Introduction

to Singularly

Perturbed

Problems

class of problems having largely an academic interest. The combined approach reviewed in the book comprises both analytical studies and a numerical analysis which complement each other. Contrary to the expansion techniques, the analytical studies in this approach are confined to investigations of qualitative properties of solutions, in particular, to obtaining appropriate estimates of the solution derivatives in the vicinity of the layers of singularities. These estimates are used to define layer-damping coordinate transformations eliminating or mitigating solution singularities or to generate layer-resolving grids with specified laws of the nodal distribution in the layers. Then an approximate solution to the problem of interest is obtained by conventional numerical techniques through the computation of the algebraic equations approximating either the transformed problem derived by the coordinate system on a uniform grid in the new computational domain or the original problem in the physical domain on the nonuniform grid whose nodal clustering in the layers is prescribed by the coordinate transformation. This approach has a definite advantage in comparison with the classical analytic expansion and numerical methods since it combines the advantages of their techniques. The qualitative analysis of solutions confined to obtaining appropriate estimates of the solution derivatives is much easier then the process of computing uniformly convergent asymptotic expansions since it allows one to reduce the investigation of the original complex multidimensional problems to simpler ordinary ones (amenable to analytical studies) with the same type of the qualitative behavior of the solution in the layers. Therefore such a cooperation of the analytical and numerical approaches will lead to the treatment of more real practical problems with singularities which do not readily succumb to the pure asymptotic or numerical methods. The approach advocated in the book is largely aimed at applications to nonlinear and bisingular equations which are not easily treated by the separate asymptotic expansions or numerical methods. The bisingular equations are the ones whose solutions have complement singularities related to the reduced (s = 0) problem. The solutions of such equations have versatile forms of layers, of course the well-known exponential ones, but overwhelmingly power and combined layers. These layers appear, for example, in the solutions of singularly perturbed equations of the second order whose coefficients before the first derivatives are not separated from zero (equations with turning points). Such a situation is common for convection-diffusion equations and for the momentum conservation equations of viscous fluid dynamics.

1.2.

Examples

of Singularly

Perturbed

Problems

3

The monograph presents some achievmentes of the combined approach developed chiefly by the author for obtaining approximate solutions. The approach is largely aimed at attacking the singularly perturbed problems with turning points which present considerable practical and academic interest. This approach is rather easy to use and ready for computer implementation and, therefore, more accessible to the practitioners or applied mathematicians. The major tools used in the book to obtain the necessary analytical results concerning estimates of the solution derivatives in the layers of singularities of nonlinear singularly perturbed and bisingular equations are the theory of barrier functions, inverse monotone operators, and comparison theorems. The barrier functions playing a role of majorants of the solution derivatives are defined through estimates of the solution derivatives of the reference equations with simple coefficients (constants or linear functions) which describe qualitatively the solution behavior of the original problem. The estimates of the solution derivatives are used to classify possible singularities, to build layer-damping coordinate transformations and to formulate appropriate laws of the grid node distribution in the layers for the purpose of defining e-uniform convergent algorithms for the numerical solution of the equations with singularities of various types which occur in practical problems. They also give some comprehension about the laws which depending on the invariants of the singularly perturbed equations control the type of the layers and their size and locations.

1.2.

Examples of Singularly Perturbed Problems

This section presents some boundary value problems for ordinary and partial differential equations with a small parameter. 1.2.1.

Convection-Diffusion Problem

Some singularly perturbed equations arise as models of convection-diffusion processes. A typical simulation of such a process is carried out through the boundary value problem with a small parameter e multiplying the n

Laplace operator V 2 = ^

Q2

—^ • 1 °xi

- e V 2 u + b(x, u) • Vu + f(x, u) = 0 ,

x G Xn ,

4

1. Introduction

u(x, er) = 0 ,

to Singularly

x G

dXn

Perturbed

Problems

.

(1.1)

Here Xn is a bounded domain, b ( x , u) is the convection vector, f ( x ) a specified vector function, while e > 0 is the coefficient of diffusion which may be small compared to b ( x , u). It is commonly assumed for the analysis of layers t h a t there is safficient compatability of the equation coefficients and boundary conditions at boundary corners of Xn so t h a t the solution of the problem has no singularities in the vicinity of the boundary corners caused by any incompatability.

1.2.2.

Momentum Conservation Laws

Some spectacular patterns of equations with a small parameter are derived from the Navier-Stokes equations. In particular, the stationary momentum conservation laws of incompressible gas flows, expressed as

f-i

dx3

j=i

dxl

¿-'dx^dxj

H

i=i

'

'

, J

'

'

v

where ul is the ith Cartesian component of the vector of fluid velocity u, p the density, p the pressure, and p, the viscosity, are singularly perturbed equations when p is a small constant.

1.2.3.

Prandtl Equations

Steady equations of Prandtl in two dimensions are presented by the following system d2u uy du

du

du

1 dp

0 + v oy + ua~ox + ~7T p ox = ' dv

,

.

where u and v are the components of the vector of the fluid velocity. If we consider the variable x as a parameter then the system is transformed t o a linear singularly perturbed ordinary differential equation with respect to the independent variable y and dependent variable u -eu"

+ v{y)u'

- v'(y)u

+ f(y)

= 0 ,

f(y)

= ^ ^ ( v )

•

When the condition of adhesion at the boundary y = 0 is imposed on the flow, then u(0) = 0, so (1.4) in this case is the equation with the

1.2.

Examples

of Singularly

Perturbed

5

Problems

boundary turning point. It is shown in Chap. 3 t h a t the qualitative behavior of u(y,e) in the vicinity of the boundary turning point y = 0 is largely dependent on the sign of u'(0). The positive sign of i/(0) means t h a t the flow is detached from the boundary while the negative sign of u'(0) is an indication of local attraction of the flow to the boundary. In consequence the turning point y = 0 is called a repulsive point if i/(0) > 0 and an attractive point if u'(0) < 0.

1.2.4.

P r o b l e m of a thin B e a m

A form of the thin beam with fixed end points is modeled by a singular perturbation two-point boundary value problem for an equation of the fourth order d4u

d2u

=

(L4)

«(0) = ti(l) = «'(0) = «'(1) = 0 .

1.2.5.

P r o b l e m of the Shock Wave Structure

One model singularly perturbed system of ordinary differential equations to investigate the qualitative structure of solutions in boundary and interior layers is represented by a problem simulating the shock wave structure of a steady heat-conducting gas flow

ax

du

pu—

da;

+

dp da:

de

du

^ da:

^da:

d2u

- = 0, ax* /du\2

^^da;^

(p u e)(0,e) = (p0 u0 e 0 ) ,

0 < d /

x
c > 0. The qualitative analysis shows that the solution can have only exponential boundary and interior layers. The position of the interior layer is computed exactly and it is determined by the boundary values Ao and A\ and solutions of the reduced problem.

1.2.7.

One—Dimensional Steady Reaction—Diffusion—Convection Model

A more general problem than (1.6) is represented by the following singularly perturbed quasilinear boundary value problem -eu"

+ g{x, u)u' + f ( x , u) = 0 ,

u(0,e) = A 0 ,

0 < x 0 is characterized by with the following types of boundary layers: exponential, two types of power layers, and combined layers, depending on the values of a(y), a'(y), and fu(y,u)/a'(y) at the boundary points, while the interior layers of u(y,e) are always of the power type.

1.3.

1.3.2.

Idealized

9

Problems

Weakly-Coupled Systems of Ordinary Differential Equations

An application of the method of lines to boundary value problems allows one to convert multidimensional scalar problems of the type (1.1) or (1.2) into two-point boundary value problems for weakly-coupled systems of ordinary differential equations - e u " + a(x, u ) u ' + f ( x , u, e) = 0 , u(a) = A ,

a < x < b ,

u(b) = B ,

(1.12)

where a ( i , u ) is a diagonal matrix. This system is obtained when one independent variable x (preferably the one along which a solution has layers) is fixed while all of the derivatives with respect to other variables are substituted by differences. A qualitative analysis of problem (1.12) with a.(x, u) = a(x) is carried out in Chap. 5.

1.3.3.

Autonomous Equation

The problem (1.5) can be simplified, for example, by the next considerations. In the case e = cvT , p = (f — 1 )pe we get from the system (1.5) -eu"

+ c[u + ( v -

1 )e/u]'

= 0 ,

- ( £ i e ' ) ' + c ( e - y + % / ) ' = (), «(0)

,

= u0

0
i{x,e) ,

e)

0 < a < 1,

ln(l + x/e) lnl/e

0 < x < 1 . (1.24)

Such m a p p i n g s are realized as solutions t o (1.21) with 1 > o(0) > 0. In this case a = 1 — a ( 0 ) . For example, when in (1.21) cf>(x,e) = 0, a(x) a. 1 > a > 0, u(0) = 0, and u ( l ) = 1, t h e n we find

A p a t t e r n of this function for a = 3 / 4 is depicted in Fig. 1.2

{right).

L o g a r i t h m i c F u n c t i o n s . O n e more set of singular f u n c t i o n s of order 1 a t t h e point x = 0 is presented by t h e m a p p i n g s u4(x,£)

l n ( £ -(- J;) = 0(x,e)———

+ (t>i(x,e),

(1.25)

in which t h e singularity is caused by t h e reference logarithmic singular function ln(e + x). T h e mappings (1.25) are obtained as solutions t o problem (1.21) with a(0) = 1. For instance, if (x,e) = 0, a(x) = 1, uq = 0, and «1 = 1, t h e n we obtain

T h e f u n c t i o n of this t y p e is shown in Fig. 1.3. S i n g u l a r F u n c t i o n s w i t h I n t e r i o r C r i t i c a l P o i n t s . T h e singular f u n c t i o n s (1.18), (1.20), (1.24), and (1.25) of order 1 having t h e b o u n d a r y point XQ = 0 as their critical point derive more general singular f u n c t i o n s of order 1 with an a r b i t r a r y critical point XQ

16

1. Introduction

Problems

Ui(x,e)

= o(x,e) e x p [ - ( x - x0)2n/ea]

+ \{x,e) ,

(l-2?)

u2{x,e)

= 4>0{x,e)ea/[e

+ ^(x^)

(1.28)

u3{x,e) u4{x,

to Singularly Perturbed

=

+ e~a)

+ cf>1 {x, e) ,

(1-29) (1.30)

where n > 1 is an integer, ex 0, 0 < ¡3 0 such that ( 1 ) the value of the function u(x,e) and its derivatives is small outside the vicinity of the point xo, specifically max |uO(:r,£)| —> 0 with e 0, 0 < I < p, for arbitrary m i , m>|x—xo|>mi m > mi > 0; (2)

max

|uW(a:,e)|

0 with £

0 for 0 < / < jfe - 1;

\x-xQ\ I > k.

The point XQ for which the conditions (1) - (4) are held is called a critical point of order k of the function u(x,e). A function u(x,e) is referred to as a function of the layer-type if there is at least one critical point XQ G [0,1] for which the conditions (1) - (4) are observed. We readily see that the functions of the layer-type are the singular functions. Contrary, if any singular mapping satisfies the condition (1) of the local concentration of its value and the value of its derivatives then it is a function of the layer-type as well. In particular the singular functions (1.18), (1.20), and (1.25) with i(x,e) = 0 are the functions of the layertype of order 1 at the point x = 0. Typical samples of the layer functions of order k at the point x = XQ are the following ones

17

1.5. Notion of Layers u(x,e)

= ek

1

exp[-(z -

X0)2/e2]

,

u{x, e) = e («+*-i)/2[6. + ( x - z 0 ) 2 ] a / 2 , u(x,e)

= ek~l

ln[e 2 + {x - x0)2]/

a > 0 ,

In e .

While the singular function of order 1 at the point x = XQ u{x,e)

= [ea + (x - x0)2f

,

0 < s < l ,

0 < a ,

0 < / ? < 1/2 ,

derived from (1.29), is not the layer-type function since there is not a point in a vicinity of which the values of the function u(x,e) and of its derivatives are concentrated, i.e. the condition (1) cannot be satisfied.

1.5.

Notion of Layers

This section gives some introduction to the notion of layers applied to singular functions.

1.5.1.

Definition of Layers

Let u(x,s) be a singular function of order A; at a critical point XQ and m be a constant specified in the definition of this singular function. A single layer of order I of the function u(x,e) at the point XQ is an interval A i 0 = [xi(e), 12(e)] C [so - m,x0

+ m]r\ [0,1]

such that 1 ) lim max e->0 |x-x0|-0xe[xi(£),X2(£)]

2) Mi > [^¿(e),e]| > M2 for some Mx > 0 and M2 for i = 1 and/or i = 2, is an interior point of [0,1];

forsome m i ,

> 0, if x f ( e ) ,

3 ) there is no interior point x3(e) of such that the conditions (1) and (2) are satisfied both on [xi(e), 13(e)] and [13(e), 22(e)]Note, if u(x,e) is a function of the layer type at xo then, according to the condition (1) in the definition of the function of the layer type, lim Xi(e) = x0 ,

i — 1,2.

T h e layer Alx = [zi(e), 22(e)] is referred to as a boundary layer if 21(e) or 22(e) is a boundary point for a sufficiently small e.

18

1. Introduction

to Singularly

Perturbed

Problems

If neither X\(E) nor x 2 (s) is a boundary point for a sufficiently small e then the layer is called an interior layer. Two different single layers having one critical point XQ are referred to as contiguous layers.

1.5.2.

Examples of Layers

The exponential function u(x,e) = e x p ( - a ; 2 / e ) ,

0< x < 1

is a function of the layer-type of order 1 at x = 0. Since u'(x,e) =

2x

exp(-2r 2 /£) ,

0 < x < 1,

the layer of order 0 at this point is Ag = [ m e 1 / 2 , ! 2 ( e ) ] where 12(e) is the number satisfying the equation u'(x,e) = —1. From this equation we obtain t h a t P\Je\ l n e | > x2(e) > \Je| Iner|/2 ,

p>

l/y/2

.

This example shows t h a t the point x0 at which a layer is formed may not be contained in the layer. For the exponential function u(x,e)

= ex.p(—x/e)

,

0 < x < 1 ,

of the layer-type of order 1 at the point x = 0, we have u(1+n*>(x,e)

= (-l)1+ne-(1+ri)

exp(—x/e)

,

0 < x < l ,

therefore, Ag = [0, (n + l ) e | lne|]. Analogously for the power function u(x,e)

=£a/(£a

+ x) ,

0 < x < 1

of the layer-type of order 1 at the point x = 0 we find U^1+n\x,£)

= (~l)n+1(n a

and consequently Ag = [0, me l^ The logarithmic function

l+2

+ l)\£a/(£a+

'>].

x)n+2

,

1.5. Notion

u(x,e)

of

19

Layers

— l n ( £ + x)/\ne

,

0 < x < 1

of the layer-type of order 1 at the point x = 0 yields 1

= (-l)"n!

(e + x ) n + 1 l n e '

therefore, Aft = [0, ra| l n e | - 1 / ( " + 1 ) ] . A function 2 «(®>e) = exp[(a; r7 - x )/£\ rTT~i+ exp[(x r? 0 0

w l ' ®o G (0,1) , 0 < x < 1 , x)/e\

of the layer-type of order 1 at the interior point zo has two interior contiguous layers of order 0 at x = XQ. Let A° 0 _ (A° 0 + ) be the layer on the left-hand (right-hand) side of XQ. Then, for this function u(x,e), we have A° 0 _ = [ ® o - e | l n e | , a ; o - m e ]

and

A ° 0 + = [zo + me, x0 + e\lne|]

.

In all the examples considered above AS0+1DA£0

and

limA"

=z0.

However for the power singular function u(x, e) = (e + x)a

,

0 < x < l , ' 0 < a < l ,

of order 1 at the point i = 0 w e find u^(x,e)

= a { a - 1) . . . (a - n + l)(e

+ x)a~n

,

therefore, Aq = [0,m], This example demonstrates the layer t h a t does not tend to the point at which it is formed and whose length does not tend to zero when e approaches 0. Other patterns of singular functions of order 1 with contiguous interior layers of order 0 are exemplified by the functions (1-27) - (1.30). Note that the functions u(x,e) and 4>o(x,£)u(x,e) + \(x,e), where \ satisfy (1.17), have the same layers at a critical point x$. Therefore for the analysis of the interior layers of the functions (1.27) - (1.28) we can assume o{x,e) = 1 and (f>i(x,e) = 0. In this case we obtain u\{x,£)

= - 2 n ( x - xo)2n~1e~a

u'2(x,e)

= —2na£a(x

U'3{X,£)

= (32n(x - xQ)2n-l[£a

- z0)2n_1/[e

2n(x U'4(X,£)

exp[—(x - z 0 ) 2 i l / e a ] ,

-

+

- z0)2n]a+1

+ {x - z0)2nF_1 z0)2n_1

ln(l + £ - a ) [ l + (x - xo)2n]/£a

'

, ,

20

1. Introduction

to Singularly

Perturbed

Problems

Accordingly, for the functions u'^x^e), i = 1 , 2 , 3 , 4 , we find t h e corresponding left-hand A° X q _, i = 1 , 2 , 3 , 4 , and right-hand A 2 r o + , i = 1 , 2 , 3 , 4 , layers as follows: A°lxo_

= [x

0

-x(e),

s0-£»/(*»)],

(1.31) A ? , 0 + = [®o + e a / ( 2 n ) , x0 +

x(e)},

where x(e) is some point satisfying t h e relation b(ea\In

el)1^2") >

, b > [a/(2n)]1^ ;

> [aea\\ne\/{2n)}ll^

A L 0 - = [ ^ o - e a / ( 1 + 2 n a ) , s 0 - e1/(2n)] , A L 0 + = [*o + e

1/(2n)

, zo + W

ALo_ = [ * o - m , A ° X 0 + = [*o + ^

(1+2na)

= [*o + e

/(2n)

1.5.3.

,

];

z0-ea/(2n)], , xo +

A ^ o - = [®o - l / l n ( l + £ — ) , x 0 1/(2n)

(1.32)

(1.33)

H ; £

1

/(2n) ]

t

(1.34)

+ 1 / l n ( l + e-0,

/3>0,

x e AlXo .

A layer A1 is referred to as a power layer when the respective derivative of the singular function is not estimated by an exponential function, however, it is estimated by the power function Mg(e)/(£a+\x-x0\)iS

,

a ,/3 > 0 .

An example of the exponential layer is given by the function (1.27), while the examples of the power layers are presented by the layers of the functions (1.28) - (1.30). A layer Alx is referred to as a combined layer when the respective derivative in the layer is estimated by a linear combination of the exponential a n d / o r power functions and is not estimated by a single one.

Classification via the Limit Behavior T h e interior and contiguous layers of a singular function u(x,e) are also classified according to the behavior of the limited function into the following three types:

22

1. Introduction to Singularly Perturbed

Problems

Figure 1.4. Functions with a single monotone layer (left) and contiguous monotone layers (right)

1. monotone layers; 2. spike layers; 3. nonmonotone layers. This division is indicated by the value of u(xo, e) at the point zo (near which the single or contiguous layers are formed) and the constants u-(x0)

= lims-^o-lime^ottO^e)

u+(i0)

= limj;_).a;0+ lim^o

, u(x,e)

.

M o n o t o n e Interior Layers

We say that a singular function at x0 € [0,1], if min{u_(z 0 ), w+(zo)}
c(x) > 0, i,j = l , . . . , n , ( x , u ) 6 [ 0 , a ] x jfr

3

. the operator

the vector-valued

T = ( L , T ) is inverse-monotone functions

in the sense

of (2.8)

u(z) £ C^Oja].

P r o o f . Let Ui(:r) = (un(x),...,

Uin(x))

a n d u 2 (a;) = (u21(x),...,

u2n(x))

be two smooth vector-valued functions in the interval [0,a] and ui(0) > U2(0) but the inequality Ui(x) > 112(2:), x £ [0,a] is not held. Then the number p =

l T[u2] results in the inequality Ui(z) > u 2 ( x ) , x £ [0,a]. So the assertion of the theorem is true.

47

2.3. Theorems of Inverse Monotonicity

2.3.2.

Second Order Equations

Scalar T w o - P o i n t B o u n d a r y Value P r o b l e m s This subsection reviews the principle of inverse monotonicity for the following quasilinear two-point boundary value problem L[u] = -[e + d{x)]'u" + f ( x , u, u') = 0 , 0 < £< 1, r[u] EE [u{0,e),u(l,e)}

d(x) > 0 , = {Ao,A1)

/ > 0, .

(2.12)

Since this single problem represents all important idealized problems (1.10), (1.14), and (1.15) modeling the qualitative features of solutions in layers, we give a detailed proof of the theorems formulated below. T h e o r e m 2.3.5. Let d(x) and f(x,u,u') be continuous in [0,1] and [0,1] X R2, respectively. Then the operator T = ( £ , r ) for the functions from C2(0,1) fl C[0,1] is inverse-monotone if one of the following conditions imposed on f is satisfied: (1)

f(x,u,u')

is strictly

increasing

in u, i.e.

f(x,Ui,z)

< f(x,u2,z)

if

ui < u2; (2) f ( x , u, u') is weakly increasing in u and there exists a constant c > 0 such that \ f ( x , u, zi) — f ( x , u, z2)\ < c\z\ — z21 . P r o o f . For the proof of the theorem we assume that there are two functions u\{x) and ^2(2) such that T[ui] < T[uj\ but the inequality u\(x) < U2(x) is false. We readily come to a contradiction in the first case, since at a point of a negative minimum of «2(2) — u\(x) we have U2(x) - ui(x)

< 0 ,

u'2(x) = u'^x)

,

«2(2:) > u'{{x) ,

and consequently L[ui](x) -

L[U2](X)

> f[x,ui(x),u[(x)]

- f[x,u2{x),u'2(x)]

>0

at this point. Now we consider the second case. The proof of the theorem in this case relies on the following observation which is valid for arbitrary function v{x) e C2(a, b) n C[a, b]. Namely, if u(a) > 0 ,

v(b) > 0 ,

min v(x) < 0 ,

x G [a, b] ,

then, for arbitrary number N > 0, there exists a point xo E {a,b) such that

48

2. Background for Qualitative Analysis

v{xo) 0 .

(2.13)

Note t h a t (2.13) is obvious when there is a point xo of a local minimum of v(x) at which v(xo) < 0 and V"(XQ) > 0. Therefore t h e relation ( 2 . 1 3 ) must be shown only for the case v" — 0 at each point of a negative minimum of v(x). Let —2m < 0 be an absolute minimum of v(x) and H be a set of the points of [0,1] at which v(x) < —m. Since u(0) > 0 and u ( l ) > 0) a n d v(x) is continuous we get t h a t H is a c o m p a c t subset of ( 0 , 1 ) . Now, in order to demonstrate the validity of ( 2 . 1 3 ) , we introduce the following mapping p(x)

= m[ 1 — exp(—bx)] ,

b> 0 .

T h e function u ( x ) + p(x) has an absolute minimum at some point XQ in the interior of H, since on the boundary of H we have v(x) = —m, 0 > v(x) + p(x) > —TO,while at a point x\, where v(x) = —2m, we get v(x\) + p ( x i ) < —TO. At the point xo we obtain v(x0)

+ p(x0)

< 0 ,

v'(x0)

+ p'(x0)

= 0 ,

v"{x0)

+ p"{x0)

> 0 ,

(see F i g . 2 . 2 ) . Hence «(zo) < 0 ,

v'(xo)

= -mbexp(-bxo)

,

V"(XQ) > mb2 exp(—bxo) .

T h i s readily yields the relation (2.13), assuming b > N. (2.13) could be proved with the choice of p(x) in the form p(x) = m e x p ( — b x ) ,

Analogously,

b> 0 .

Now, turning to the proof of the theorem, we write out t h e following relation for v(x) = « 2 ( 2 ) — u\(x) at any point x

F i g u r e 2.2. Illustration to Theorem 2.3.5

2.3.

L{u2){x)

Theorems

- Llu^x)

=

of Inverse

d{x)]lv"{x)

~[£ +

+f[x,u2(x),u'2(x)] =

49

Monotonicity

-

f[X,iti(z),u'i(z)]

+f[x,u2(x),u'2(x)]

-

f[x,u!(x),u'2(x)]

+f[x,u1(x),u'2(x)]

- /[x,

d(x)}!v"{x)

~[£ +

.

If the inequality u2{x) > u\(x), 0 < x < 1 is false, then at a point XQ at which (2.13) is satisfied for v(x) with a sufficiently large N, we readily obtain from the above expression for L[u2](x) — (x) that L[u2](xo) < L\U\\{XQ) which contradicts to the assumption L[u2] > L[ui], So the theorem is true. Similarly to Corollary 2.3.1 we also have the following corollary. Corollary 2.3.2. Let f ( x , u , z ) = b(x)g(x,u, d(x)

and

f ( x , u , z ) be continuous

g(x,u,z)

be of class

let there gu{x,

exist

C1

a positive

u, 0 ) > c ( x ) > 0 for

problem

( 2 . 1 2 ) is limited

with

z)

respect

continuous

to u for function

(x, u) in [ 0 , 1 ] X R. by the following

with

b(x)

> 0, and

[ 0 , 1 ] X R2,

in [ 0 , 1 ] and

let

respectively,

£ [ 0 , 1 ] X R2,

and

c(ar) > 0 on [ 0 , 1 ] such

that

(x,u,z) Then

the solution

u(x,e)

to

bound

| u ( z , e ) | < M = max{ max \g(x, 0,0)|/c(x), |A 0 |, lA^} , 0 < x < 1 . x6[0,l] (2.14) This estimate follows from the obvious implication for the inversemonotone pair T = (L, T) T[-M]

< T[u]

< T[M]

-M

< u{x,e)

< M ,

0 < x < 1 ,

which is valid in accordance with Theorem 2.3.5. Now we consider one specific representation of problem (2.12) in the following form: L[u]

=

-[e

+ d(x)]lu"

0 < e < 1,

+ g(x, d{x)

u)u' + f ( x , u ) = 0 , > 0 ,

TO

T[u] = [t t (0,e),ti(l,e)] = ( ^ o M i ) •

0 < a; < 1 ,

> 0 ,

(2-15)

Physical explanations and justifications of this problem for d{x) = 0 are given in Sect. 1.2.7. In the formulation of Theorem 2.3.5 we assumed a rather tough restriction of a strong or a weak monotonocity of f(x,u,u') with respect to u which, in application to (2.15) with smooth functions g(x,u) and f(x,u), means that

50

2. Background

gu(x,u)u'

+ fu{x,u)

for

> 0 ,

Qualitative

or

gu(x,

Analysis

u)u'

+ fu(x,

u) > 0 ,

and therefore depends on u'(x,e). Here we consider a more suitable restriction which is adjusted to the form of problem (2.15) and does not depend on u'(x,e). T h e o r e m 2.3.6. Let in (2.15) d(x) G R),

and

0,1 ],gu, gx, and fu G C([0, l ] x

either

(1)

g{x,u)

= b(x)

/U(s,u)>0,

,

G (0,1) x

{x,u)

R ,

or ( 2 ) (wu

- ax)(x,

u)>

0 ,

( x , u) G [ 0 , 1 ] X R ,

where w(x,u) Then

= f ( x , u ) / [ e + d(x)]1,

the operator

ity property

(L, T)

T =

( 2 . 8 ) for

from

the functions

a(x,u)

(2.15) u(x)

+ d(x)]!

— g(x,u)/[e

satisfies

the inverse

belonging

to C2{0,1)

.

monotonicn

C[0,1].

Proof. The first case is readily turned to the restriction (2) of Theorem 2.3.5. Note that in this case the conditions of the theorem can be mitigated turning to the following ones: d(x) and b(x) G C[0,1]

and

fu(x, u) G C[0,1] X R .

Let us now consider the second condition. We assume that ui(:r) and U2(x) are arbitrary C 2 (0,1) fl C[0, l]-functions for which T[ui] < r[u 2 ]

and

L[ui]

< L[u2]

,

0
0, the pair (B*, T) is inverse-monotone, in accordance with the first case of Theorem 2.3.6, therefore it has a corresponding nonnegative Green's function G(x,£). If we now assume that the inequality u\(x) < U2{x) is not valid in [0,1] then, taking advantage of the condition r[ui] < r[u2], we obtain that there is an interval x ( z 0 , z i ) £ [a, o < such that v(x) < 0 for xq < x < x\. Let r(x) 6 C°°[2;o)^i] be a nonnegative, nonzero function satisfying the conditions Ak\x) = 0, k = 0 , 1 , . . . , for x < xq and x > x± and Vi(x) be a solution of the boundary value problem B*(vi)

The function

= r(x),

0 0 if g(x,u) does not depend on u or if d = 0 and g(x,u) does not depend on x.

Corollary 2.3.3. the following

Letu(x,e)

be the solution

/(:r, u) = b{x)r{x,

u) ,

b(x) > 0 ,

r(x, u) and g(x, u) 6 C 1 , 1 ( [ 0 , 1 ] X where

c(x)

of {2.15)

whose

terms

satisfy

conditions:

is a continuous

positive

d(x) and b(x) £ C [ 0 , 1 ] , ,

function

and ru(x, on [ 0 , 1 ] .

u) > c(x) > 0 , Then

|u(z,£)| < M = m a x { max |r(ar, 0)|/c(x), |A0|,

, 0 < x < 1 .

x6[0,l]

(2.16)

P r o o f . For the proof of the corollary we introduce the operator as follows: Lu[v]

= -[e

+ d{x)]mv"

+ g[x, u{x,e)]v'

+ f(x,

v) .

Availing us of Theorem 2.3.6 we conclude that the pair ( L u , T) is inversemonotone. We also have the following obvious inequalities LU[M]

> 0 > Lu[-M]

,

r [ M ] > r[u] > r [ - M ] .

Since Lu[u] = 0, we obtain from the inverse monotonicity of ( L u , T) t h a t \u(x,e)\ m > 0 for (x,u) G [0,1] X [—M,M] then the solution u(x,e) to (2.15) does not have interior layers of large variation and consequently may have only boundary layers. Really, if u(x,e) had an interior layer of large variation this would mean that the function u'(x,£) has a local extremum at a point £ 0 = xo(e) € ( 0 , 1 ) , and consequently u"(x0) = 0. B u t then we obtain from (2.15) and the condition |g(x, u)| > m > 0 that

2.3.

u'(x0,e)

Theorems

of Inverse

Monotonicity

< \b[xo,u(x,£)]/a[xQ,u(x,£)]\

53

< M\ .

Thus the first derivative may tends to infinity only near the boundary points when e approaches zero. So there is valid the following corollary. C o r o l l a r y 2 . 3 . 4 . The solution u(x,e) of (2.15) does not have interior layers if the terms of (2.15) satisfy the conditions of Corollary 2.3.3 and, besides this, \g{x, u)| > m > 0, for x G [0,1] X [-M, M], where M is the constant bounding |ii(x,e)| for all e, 0 < e < m i . S y s t e m s of T w o - P o i n t B o u n d a r y V a l u e P r o b l e m s The principle of the inverse monotonicity is also valid for two-point boundary value problem (2.7) with some restrictions on its terms. T h e o r e m 2 . 3 . 7 . Let in (2.7) i{x, u, u') = A ( z ) u ' + g ( z , u), where A ( x ) = {¿ja,(a;)}, i, j = 1 , . . . , n, is a diagonal matrix, g(x, u) = [g\{x, u ) , . . . , gn{x, u)] is a smooth vector-valued function with respect to ( z , u ) G [0,1] x Rn. Let also (1)

(x, u) G [0,1] X Rn ,

u) < 0, i + j, i,j = 1 , . . . , n,

1

3 j^i n ( x , u ) € [0, a] x R ,

where c(x) is a continuous function (L, T) from (2.7) is inverse-monotone u(x,e) G C 2 [0,1],

in [0,1]. Then the operator T for the vector-valued functions

=

P r o o f . Let two vector functions u i ( i , e) = [un(x, e),..., uin(x,e)] and 2 U2(x,e) = [ u 2 i ( x , e ) , . . .,U2n{x,£)\ belong to C [0,1] with respect to x and let T f u j ] > T[u2]. We designate by p the following constant p=

min [uu{x,e) l L[u 2 ] and T[ui] > r[u 2 ]. This completes the proof of the theorem. Elliptic E q u a t i o n s We also consider nonlinear elliptic differential equations of the form n

L[u] = -€

+

ff(x>

ux) = 0 ,

xeG.

(2.17)

i,i=1 Here G is an open bounded domain in Rn, u x is the gradient vector of u, i.e. u x =

•••

and

{ ^ ( x ) } i h3

=

1, • • •, n i is

a

uniformly

positive C(G) matrix, i.e. there exists a constant c > 0 such that n ij = 1

M x ) C £ j > cllill 2

for all

i e r ,

xeG.

T h e o r e m 2 . 3 . 8 . Let 0 , xi < x < x2. Analogously sgn(ii') = - 1 , if u'(x,e) that

< 0,

u"{xi,e)

Xi < x < x2.


m > 0, k > 0, and a(x) G C[0,1]; £~m exp(-xk

+

< M[ 1 + £~m exp{-xk/£k)]

,

0 < x < m1 ,

(2.39) for a > m > 0, k > 0, a(x) £ C[0,1] and some mi > 0.

2.6.

Comments

Differential inequalities as a suitable tool to carry out a qualitative analysis of solutions to differential equations were first applied by Peano (1885/1886), Bernstein (1912), Perron (1915), Nagumo (1939), and Chaplygin (1950). In particular, Nagumo conditions were in fact formulated and applied by Bernstein (1912). Nagumo (1939) was the first mathematician who used differential inequalities to analyze the initial-value singularly perturbed problem

64

2. Background

EU" = f(x,

-u(O) =- uq ,

for Qualitative

u,u',e)

,

Analysis

0 < x < A ,

= u'q .

(2.40)

The stronger version of Nagumo's theorem with a generalization to nonsmooth barrier functions was proved in Jackson (1968) and Bernfeld and Lakshmikantham (1974). The application of Nagumo's theory for obtaining global solution and derivative estimates arising in problem (2.7) with £ = 1 was demonstrated by Hartman (1960). The Nagumo theory was also extended to arbitrary mixed boundary problems by Heidel (1974). The Nagumo results were applied by Brish (1954) to study a two point mixed boundary problem with the equation from (2.40). In particular, he proved the existence of the exponential type boundary layer if |/u,| > m > 0. The existence and asymptotic behavior as e —> 0 of solutions of problem (2.7) were studied by Kelly (1979, 1984) using differential inequality techniques. The theory of differential inequalities has been used in a number of applications (see, for example, Schroder (1980) and Bernfeld and Lakshmikantham (1974)). Theorems 2.3.6 and 2.4.1 for d{x) = 0 were proved by Lorenz (1982). An estimate similar to (2.26) was proven in the discrete case by Abrahamsson and Osher (1981). Theorem 2.4.1 for d(x) > 0 was proven by Liseikin and published in the monograph by Liseikin and Petrenko (1989) and in Liseikin (1992a). A more detailed description of the most results concerning the inequality theorems presented in this chapter can be found in Walter (1970) and Chang and Howes (1984).

Chapter 3

Estimates of the Solution Derivatives to Semilinear Problems 3.1.

Introduction

This chapter establishes some estimates of the solution derivatives to initial and two-point boundary value problems for ordinary semilinear singularly perturbed equations with a small parameter e. Such equations model qualitative properties of convection-diffusion processes or movement of gas flow near boundary segments or around holes of a small diameter. Solutions of these equations depend on a variable x and the small parameter e and are designated by u(x, e). We are interested in obtaining the estimates of the derivatives of u(x,e) with respect to x. In the sequel, we shall adhere to the following notational convention for designating the derivatives as U'(X,E), U"(X,E), U"'(X,E),OT U^(X,E). An important role in formulas for the estimates belongs to basic singular functions delineated in C h a p . l and to constants which are the same for all values of e from the interval (0,1]. Remind t h a t these constants are typically designated by M , m, M{, and mj. The chapter is concerned with an analysis of the equations having the small parameter e in a coefficient before the higher derivative only. However, the results are readily extended to the problems whose terms and boundary d a t a depend on e as well but in a regular fashion. Proper modifications of the theorems proved are readily formulated.

3.2. 3.2.1.

Initial Problem Smooth Terms

In this section we study the following initial-value problem L[u] = eu' + f(x,u)

= 0,

z > 0 ,

r[u] = u(0,e) = Ao , with f(x,u)

£ C n ' n ( [ 0 , a] x R), n>

1> e > 0, (3.1)

1.

66

3. Estimates

of the Solution

Derivatives

to Semilinear

Problems

T h e o r e m 3.2.1. Let u(x,e) be a solution of (3.1). Then, for 0 < i < n and 0 < x < a, the following estimates are held | i i W ( x , e ) | < M { 1 + e~ i exp[—c(0)x/s:]} , if fu{x,u)

(3.2)

> c(x) > 0 where c(x) € C[0, a];

|u ( i ) (s>e)l < M [ l + e - i / ( k + V e x p ( - m x k + 1 / e ) + (£ 1 H 1+ V + x ) 1 - i ] , (3.3) if f(x,u) = xkg(x,u) and gu(x,u) > c(x) > 0 where c(x) G C[0,a], 0 < m < c = min ie [ 0)O ] c(x), k > 1 is a positive integer. P r o o f . In the both cases we have, from (2.10), the following estimate for the solution u(x,e) of (3.1) \u(x,e)\(x,e)

= e -

1

f Skgu[t,u(t,e)]d£

,

o we, owing to (3.4) and (3.9), find that Lu[-b]

< Lu[g]

< Lu[b]

,

e1/(1+fc)

- b ( e l ' ^ k ) , e ) < g[elHl+k),u{£lKl+k),e)}

< x < a , < b{elKl+k\e)

,

if M in the expression for b(x,e) is sufficiently large. So, from the inverse monotonicity of the pair (Lu, Ti) where Ti(u) = u(£ 1 /( 1+ *)) (see Theorem 2.3.1), we obtain

68

3. Estimates

of the Solution Derivatives

| g[x,u{x,e)]\

< b(x,e)

,

to Semilinear

e^+k)


0, not only if it is a positive whole number. Now using (3.7) with f(x,u) = xkg(x,u) gives X

u'(x,e)

= £~l J{A**"1

{x, e)]d£.

o (3.11) Since exp [(£, e) - {x, 1. Contrary to (3.1), the second item in the equation from (3.20) is not a smooth function at the turning point x — 0. T h e o r e m 3.2.2. 0 < i < n and 0 < |uW(®,e)| where

u(x,e)


c>

x
0, then the turning point xo is referred to as a repulsive turning point.

3.3.1.

S t r o n g Ellipticity

Let us consider here the two-point boundary value problem (3.23) with a(x) fu(x,u)

e Cn[0,1]

,

f ( x , u) £ Cn'n+1

( [ 0 , 1 ] x R)

> c > 0 , f o r (x, u) e [ 0 , 1 ] x R .

,

(3.24)

The function f(x, u) with the assumption (3.24) provides the property of inverse monotonicity for the operator T = (L, T) (Theorem 2.3.5), and consequently rules out the so-called resonance cases. In accordance with Theorem 2.4.1, the total variation of the solution u(x,e) of (3.23) with the condition (3.24) for n > 1 is e-uniformly bounded thus guaranteeing the existence of a transformation x(£,s) : [0,1] —> [0,1] such t h a t the first derivative of the function tti(£,£) = u[a;(^,e),e] is also e-uniformly bounded, i.e. | d « i / d £ | < M , 0 < \ < 1. In this section we find the estimates of the derivatives of u(x, s), which allow one to generate explicitly the transformation £(£,£) t h a t eliminates the singularities of u(x,e) up to a necessary order k. These singularities may appear in the vicinity of the boundary points of the interval [0,1] and near certain subset of the set of all of the interior turning points. This subset is finite for any k > 0. We now describe the conditions to which these interior points satisfy.

3.3. Second Order

73

Equations

For n > 1 and any positive whole number k meeting the restriction 0 < k < n we denote by x\, xk, • •., all of the interior points of the interval [0,1] each of which satisfies the restrictions = 0

a(xk)

and

- ka'(xk)

> c ,

i =

1,2,....

Owing to compactness of [0,1], the number of these points is finite for each k > 0. Contrary, because of compactness of the closed interval [0,1], there would exist a point x0 such that in an arbitrary vicinity of xo the number of the points xk is infinite. Without loss of generality we can assume that this point xq is a single one and limj^oo xk = xq. Consequently we find that lim a(x'j) J j-+oo

= a(xo) = 0

— k lim a!(xkA — —ka'(xo) j-y oo

and

> c

because the function a(x) is smooth. Since a(xj) = 0, j = 1 , . . . , we conclude from the mean value theorem that in each interval there exists a point x° such that a'(x°) = 0. It is obvious that lim^oo = xo, therefore, we obtain that lim^oo a ' ( x = a'(xo) = 0, which contradicts to the above estimate — A;a'(a;o) > c. Thus for any k > 0 the number of the points x\• is finite. Let this number be designated by NkT h e o r e m 3.3.1. Let u(x,e) be the solution of (3.23) with the (3.24). Then, for 0 < k < n and 0 < x < 1,

condition

|«W(x,er)| < Af{l + 0 o [®,*r I a(O),a / (O),e] + 0i[s I Ar,o(l),a / (l),e] Nk

+

g(ei/2

|a._xJ|)«J-fc})

+

(3.25)

3=1

where each ok, j = 1 , . . . , Nk, is an arbitrary of £ and satisfying

positive

number

1

0 < a - < \c/a'(xj)\,

the restriction

independent

j =

1

,...,Nk,

while 4>i(x, k, a, b, e) = 4>o(l - x, k, —a, b, e) and

£~ k exp(ax/e) , e

6o(x,

k.a.b.e)

a/2(£l/2

+

a

.)-a-fc

a < 0, )

O = 0,

b>

0,

a = 0,

6 < 0 ,

— {e1'2

0,

+ x)a-k

+ e{e1/2

+ x)-k-2

,

a >

0,

74

3. Estimates

of the Solution

Derivatives

to Semilinear

Problems

with a independent of £ and observing the restriction 0 < a < |c/6|, in particular, if b — 0 then a is an arbitrary positive constant. P r o o f . We confine ourselves to the substantiation of estimate (3.25) only for k = 0 and k = 1, since in the case k > 1 all of the calculations are made by analogous arguments over the differential equations satisfied by these higher derivatives, which are obtained by differentiating k times the original equation in (3.23). In the process of proving Theorem 3.3.1 we consider separately all of the necessary cases indicated by the respective subsections whose outcomes will result in estimate (3.25). To prove the theorem, we first note that the operator T — (L, T) satisfies the inverse monotonicity property which holds, in accordance with Theorem 2.3.5, in the case / U ( £ , u ) > 0. Solution Value B o u n d s In view of the inverse monotonicity property of the operator T = (£,r) the solution to the boundary value problem (3.23) with the condition (3.24) is uniformly bounded with respect to e, since fu(x,u) > c > 0 (see Corollary 2.3.3), i.e. \ U { X , E ) \ < M ,

0
0, because a(x0) smooth function, we obtain

= 0 and a(x)

is a

76

3. Estimates of the Solution Derivatives to Semilinear m,x,e)\
0 if the both points £ and x lie in t h e interval (xo — me1!2, xo + me1/2). Thus, assuming x = £ i ( e ) in (3.28), we find t h a t |-ix'(xo, e)| < Me'1!2 for \x — £o| < me1!2. Using once more (3.28) gives the estimate (3.27) for \x — £o| < me1!2 if a(a?o) = 0- Differentiating (3.23) twice and resolving the resulting equation with respect t o u"(x,e) in the form of (3.28) we, by the same a r g u m e n t s as in the case k = 1, find the lower line of estimate (3.27) for k = 2 in the vicinity of t h e t u r n i n g point xq. Analogous scheme is carried out to prove (3.27) in t h e vicinity of the turning point xo for k > 2. E s t i m a t e s (3.27) are very rough and they exhibit only t h e m a x i m u m bounds on the derivatives in the whole interval [0,1] and in t h e vicinity of the turning points. We now proceed to establishing more accurate estimates. E s t i m a t e s near N o n t u r n i n g Points Let us first consider the case a(0) < 0. Then a(x) < —m, for 0 < x < m, and some m > 0 and consequently 0. Let us now assume that a(0) > 0. Then a(x) > m in the interval [0, m] for some m > 0. By virtue of the mean value theorem and estimate (3.26), there exists a point a ^ e ) in the interval [m/2, m] such that | e ) | < M . Substituting this point £o(£) for xo in (3.28) we readily obtain \u'(x,£)\
0 ,

since (£)

e)


£ > x ,

i.e. estimate (3.25) for k = 1 in the vicinity of x = 0 with a(0) > 0 is valid. In the same manner, successive differentiating of (3.23) gives \u{k)(x,£)\
0 and M > 0, such that

78 , (k),

3. Estimates

of the Solution

m^ „ /

Derivatives

to Semilinear

1 + ^ e x p h l

-

I 1,

Problems

a(l)>0, a(l) < 0 ,

3.32

for 1 — m < x < 1 and k < n + 1. Also, for each of those interior points y of [0,1] which satisfy the relation a(y) ^ 0 we can choose a point XQ which meets the inequality |w'(a;o)| < M on the left-hand side of y if a(y) < 0 and on the righthand side of y if a(y) > 0. By this, from (3.28), we readily obtain that |ii'(x,£)| < M for |a: — y\ < m with some m > 0. Analogously, after successive differentiating (3.23), we find that (x,e)\ 0. Boundary Turning Points Now we estimate u'(x,e) in the neighborhood of each boundary turning point XQ = 0 and XQ = 1, i.e. when a(x) in (3.23) satisfies the condition a(x 0 ) = 0. Case a(0) = 0, a'(0) > 0. In this case we introduce a family of operators Lui k > 1, by the following formula Lku[v] = -ev"

+ a(x)v'

+ {fu[x,u(x,e)]

(3.34)

+ ka'(x)}v

where u(x,e) is the solution to (3.23). Taking into account (3.23) and (3.26), we obtain for some M > 0 \Ll[u']\

= \fx[x,u(x,e)]\
m ,

(3.35)

ai(0) > 0, and

0< x < m .

Hence the pair (L\, T), where T is the operator of the boundary values, is inverse-monotone for the functions of class C 2 [0, m]. In order to estimate u'(x,e) we introduce the bounding function

3.3. Second Order

g(x,e)

= M[ 1 + e * / V

/ 2

79

Equations

+

,

0 < a < c/a'{0)

.

Here a is a constant independent of e. In particular, if a'(O) = 0, then a is an arbitrary positive constant satisfying the condition a > mi for some mi > 0. We have eal2 K[g]{x,e)

= Af

/ 2 +

+3gi(a,g)

+ M{f

[ x , u(®,e)]

u

+

a'(®)}

where gi{x,e)

— ( a + l ) ( a + 2 ) e - ( a + l)a{x)(ell2

=

+ a'(x)}(e1/2

+{fu[x,u(x,e)]

+

+ a;)

x)2

- ( < * + l ) ( a + 2 ) e + (a + l)e 1 / 2 a 1 (a;)(e 1 / 2 + ®)

=

(a;) + a'(a;) - a ^ a : ) } ^ 1 / 2 + a;)2 with a\{x) = a(x)/x. Note that the conditions a(0) = 0 and a(x) € 1 C [0,1] result in ai(x) G C[0,1]. Since fu(x,u) > c > aa'(0) - aci(0) and a'(0) — ai(0) = 0, there exist constants m > 0, mi > 0, mi € [0,m] and m2 > 0 such that gi(x,e) > 0 for mi > x > miE1!2 and 0 < £ < m2. Therefore > m i e 1 / 2 < a; < mi , 0 < e < m2 . (3.36) +{fu[x,u(x,e)]

- aai

Moreover, there exists a point a:i(e) in the interval [mi/2, mi] such that |u'[a;i(e),e]| < Mi. So taking a sufficiently large M in the expression for g(x,e) we obtain from (3.27), (3.35), and (3.36) Li[g] r[g]

> Ll[u'] >

>

> Li[-g] T[-g]

,

mi^/2

< ® < ^(e)

,

,

where r[u] = { ^ ( m i f 1 / 2 ) , « ^ ! ^ ) ] } . From the inverese monotonicity of the pair (L*, T) we conclude |u'(x,£)|

< g{x,e)

,

mie1/2

< x < mi/2

.

This and (3.27) result in |u'(a;,e)| < M[l + eal2{ell2

+ x)-a-x]

,

0 < x < mi/2 ,

i.e. in estimate (3.25) for k = 1 in the vicinity of the boundary point x = 0 if a(a;) and a satisfy the following restrictions

80

3. Estimates

of the Solution

a(0) = 0 ,

Derivatives

a'(0) > 0 ,

to Semilinear

and

Problems

0 < a < c/a'(0) .

In the same manner, using the operators Lk, we find the following estimate k u ( \x,e)\ 0, if the above restrictions for a(x) and a are held. Analogously we can employ the same operators Lk, k > 1 in proving estimate (3.25) for u^k\x,e) in the neighborhood of the point xo = 1 if a(l) = 0, a'(l) > 0. Thus, as (3.37), we obtain |ti (fc) (x,e)| < M[1 + ea/2(£1/2

1 - x)-a~k]

+

1 - m < x < 1 , (3.38)

,

for k < n and some m > 0, if a(l) = 0, a'(l) > 0 , 0 < a < c/a'( 1) and besides this a is independent of e. Boundary Turning Points with a'(xo) < 0. Assume now that a(0) = 0, a'(0) < 0. Then there exists a constant m > 0 such that a'(x) < —2m for 0 < x < m, and consequently {£,,x,s) 0, the estimate (3.39) is too rough for u'(x,e) accordance with (2.26), there is held the following inequality

since, in

l J |ti'(ar,e)|da: < M . o Therefore it transpires that, for the purpose of generating a layer-damping transformation, we have to improve the estimate (3.39). First we consider the case 0 < — a'(0) < c. With this assumption the pair (Li,T) where L \ is the differential operator defined by (3.34) and T is the operator of the boundary values is inverse-monotone in C 2 [0,m] for some m > 0. Now, taking the function = M { 1 + £ - 1 / 2 exp[(0, x,e)]}

g(x,e)

0 < x r M

>Li[-g] , > T[-g] ,

0 < x < m, r[W] = [«(0), «(m)] ,

which results in the estimate |ti'(x,e)| < M { l + e~ 1/2 exp[0(O,a;,e)]} ,

0< x < m ,

for some 1 > m > 0. In the same way, using successively the operators we find e)| < M{l+e~k/2exp[(f>(0,x,e)]} where m is such that c + ka'(x) and (2.39) we have |«(fc)(®,c)|

< M[l +

,

0 < x 0 for x £ [0, m]. From this estimate

e-k/2exp{-a'(0)x2/e)}

< M [ l + £ - k / 2 e x p ( - m Q x / e 1 / 2 ) ] , 0 < x < m , (3.40)

82

3. Estimates

of the Solution

Derivatives

to Semilinear

Problems

for some m > 0 and arbitrary m0 > 0, if a(0) = 0 and 0 < —A;a'(0) < c. Performing the transformation x —> 1 — x, we readily get the following estimate for u ^ ( x , e ) in the vicinity of the boundary point x = 1, if a ( l ) = 0 and 0 < - k a ' ( 1) < c, u , W ( x , e )


c, we use the preliminary estimate (3.39) valid for arbitrary a'(0) < 0. To this end, we introduce an operator Lu by the formula L

u

=

[ v ]

- E V "

+

a ( x ) v '

+

f u [ x ,

u ( x , e ) ] v

,

where u ( x , e ) is the solution to (3.23). Since f > c , it is clear that the operator ( L u , r ) , where V is the operator of the boundary values, is inverse-monotone in C 2 [ 0 , 1 ] . We have u

L

u

[ f ]

=

e [ f u u { u ' )

+

2

2 f

x

u

u '

+

f

x

]

x

+

a ( x ) f

x

.

Here / = f[x, u(x, e)], while fuu, fxu, fxx, and fx stand for the corresponding partial derivatives of f ( x , u ) at the point [x, u ( x , e)]. Taking into account (3.27), (3.39), and the condition a(0) = 0, we find that I L

u

[ f ] { x ) \
0. bounding function

1

/

+

2

x ) ~

2

+

x ]

In order to estimate

g ( x ,

e )

=

M

[

e

(

^

/

2

+

x ) ~

0
Lu[f] \f(x,

> Lu[—g] ,

u(x, e ) ) | < g(x,e)

mxe1!2

< x < mi ,

x = m\£ll2

,

and x = m i ,

provided that M > 0 in the expression for g(x,e) is sufficiently large. Consequently, by virtue of the inverse monotonicity of ( L „ , r ) , where r[u] = [v(m,\£ll2, u(mi)], we have \f[x, u(z,e)]|

< g{x,e)

m\£X^2

,

< x < m\ ,

e < m^ .

From this estimate and (3.26) we find that \f[x,u(x,£)]\ c. Let m > 0 be a constant such that a'(x) < - 2 m using for x € [0, m]. Then, taking into account the estimates (3.26), (3.27), and (3.43) in (3.28) we obtain, for 0 < x < m, |u'(a;,e)|


0, for 0 < x < 1. Therefore, the solution derivatives in the vicinity of x = 0 are estimated by (3.37), while near x = 1 they are estimated by (3.49). Since a(x) > 0, for 0 < x < 1, the solution does not have interior layers (see (3.33)). Hence in this case we conclude t h a t |«(fe)(x,e)|


—a'(xo). Then there exists a c o n s t a n t m > 0 such t h a t c > —a'(xo), for \x — xo| < M, 0 < XQ — M, XQ + m < 1 and consequently t h e pair T = where L\ is defined by (3.34) and T is the operator of t h e boundary values, is inverse-monotone in C2[xo — m, xo + rri\. Further, for every £ > 0, there exist points xi(e) and X2(e) in t h e intervals (xo — m, xq — m/2) and (x L\(-M)

M

T h u s from the inverse monotonicity of we obtain t h a t |u'(a;,£:)| < M ,

,

Xl(e)

— a'(xo)- Analogously, using t h e o p e r a t o r that lu(^(x,s)l —ka'(xo). So the estimates (3.33) and (3.51) demonstrate that the solution u(x, e) of (3.23) with the condition (3.24) may have interior singularities of order k only in the vicinity of those interior turning points xq at which —ka'(xo) > c. As was proved in the beginning of Sect. 3.3.1, the number of these points is finite for an arbitrary convection term a(x) € C^O, 1]. This very number, designated by Nk, is included in the second line of the estimate (3.25). The condition —ka'{xo) > c means that the convection term a(x) must be strongly decreasing function near this turning point xq for producing an interior layer. Thus, if a(z) ^ 0, 0 < x < 1, or a'(x) > 0 at all interior turning points, then the solution u(x,e) may have only boundary singularities. Estimates (3.31) - (3.33) and (3.51) prompt a choice of the term a(x) which results in uniform bounds in e of the solution derivatives to problem (3.23) with arbitrary boundary conditions and f(x,u) satisfying (3.24). For example, if a(0) > 0 ,

a(l) < 0 ,

and

a'(x0)

> 0

at each turning point xq then we obtain u(k)(x,s)l

< Mk ,

032 ,

|a; — £o| |a; - ®0| < e*3*'0 ,

\f[x,u(x,E)}\ 0. This estimate is represented by the second line of the general formula (3.25), for all points Xj at which —ka'{xk-)

> c. In p a r t i c u l a r , for t h e f u n c t i o n a ( x ) in F i g . 3.3 (left),

we

conclude from (3.31) - (3.33) and (3.59) that the solution u(x,£) may have only one interior layer in the vicinity of the turning point xo, while the solution derivatives are estimated by (3.59) with 0 < x < 1. A possible qualitative behavior of the solution u(x,e) is demonstrated in Fig. 3.3 (right).

General C a s e Thus we have estimated the first derivative of the solution to (3.23) with the conditions (3.24) near an arbitrary point of the interval [0,1]. By

91

3.3. Second Order Equations

a

u

o

1 X

0

1 X

Figure 3.3. Solution behavior for the equation with an interior turning point

compactness of [0,1] estimate (3.25), for k = 1, follows from these local estimates. The remaining estimates for k > 1 are proved in a similar way by sequentially differentiating (3.23) and using estimate (3.25) obtained in the previous steps. This completes the proof of Theorem 3.3.1. Examples of t h e Convection Term Deriving Various T y p e s of Singularities We now discuss some types of the convection term a(x) and possible types of the corresponding singularities of solutions. The singularities rely on the estimate (3.25) which depends on the values of a(x) at the boundary points and of a'(x) at the turning points. The patterns of the function a(x) are depicted in Fig. 3.4. Since ai(0) > 0 and a i ( l ) < 0, the function a(x) — ai(z) may produce a solution with a single interior layer containing the point x\ analogously a

a2(x)

« aA(x)

Figure 3.4. Examples of the convection term a(x) causing various types of solution singularities near boundary and interior points

92

3. Estimates

of the Solution Derivatives

to Semilinear

Problems

to the function ia(x, £r) in Fig. 3.3 (right). In the case a(x) = a2(2) the first derivative of a2(x) at the turning point x = x2 is positive, so the solution may have only two exponential boundary layers, since 02 (0) < 0 and «2(1) > 0- The derivatives are estimated in this case by the formula |uW(®,e)| < M { 1 + e~ k exp[a2(0)a;/£:] + e~ k e x p [ a 2 ( l ) ( i — l)/e]} . The solution derivatives for the example a(x) = 0,3(x) are estimated by (3.50). This solution also may have only boundary layers, however, in contrast to the case a(x) = 02(1), they are produced by power singularities. In the fourth case represented by a(x) = a^(x) the solution may have two boundary layers and one interior layer near the turning point X4. Since a'(0) > 0 and a ' ( l ) > 0 both boundary layers are of the types represented by the function of the kind (1.20) and depicted in Fig. 1.2 (left). The solution derivatives are estimated in this case by the following inequality |w(fc)(x,0|


Bu[u'\

> Bu[-g]

,

0 < x < m ,

r[ IV] > r[-g],

r[w] = Ko)X™)],

where g = M[ 1 + £~1/2

expi-yfix/e1/2)}

with M sufficiently large. Thus from the inverse monotonicity of the pair T = (¿? u ,r) we find that estimate (3.60) is true for k = 1. Analogously we obtain estimate (3.60) for k > 1. Making a transformation x —> 1 — x we readily obtain the following estimate of the form (3.60) \uW(x,e)\
0. Now introducing the bounding function = M[ 1 + ea/{e1'2

g(x,e)

+ x)a+1]

,

a = c/a'(0)

,

we obtain D

M

=

Je^Tx)^

9 l { x

e )

'

+

M [ f u + a (

' °)] '

where 9l(x,e)

=

(a+l)[a>(0)£i/2(e1/2

+ x ) - ( a +

+{fu[x,u(x,e)}-aa'(0)}(e1/2

2)e]

+ x)2

.

As a'(0) > 0 and fu > a a ' ( 0 ) , there exists some positive constant mo in the interval [0, m] such t h a t 9i{x,z)

m0£1/'2

> 0 ,

< x < m0 .

Therefore, taking into account (3.37), we find Du[g]

> Du[u']

> Du[-g]

m0e1/2

,

,

< x < m0 ,

r[v] = ^ ( m o e 1 / 2 ) , «(mo)] ,

r[flf] > T[U \ > r[-flf] ,

for M sufficiently large in the expression for g(x,e). verese monotonicity of the pair ( D u , T) we obtain \u'{x,e)

\ < g(x,e)

moe1^2

,

Thus from the in-

< x < mo .

This and (3.37) yield the following estimate \u'(x}e)\
0, and a = c/a'{0). Notice the constant a in (3.37) satisfies 0 < a < c/a'(0). Analogously we obtain an estimate of the form (3.64) in the vicinity of the boundary point x = 1, if a ( l ) = 0 and a ' ( l ) > 0,

3.3. Second Order \u(k\x,e)\ 0, for xo = 0 or xo ' 1, that < M [ 1 + £ a / 2 ( e 1 / 2 + |® - xo\)-a~k]

,

0 < \x - ®0| < m (3.66)

with 0 < a < c(xo)/a'(xo). Also if a(xo) = 0, a'(a;o) < 0, for xo = 0 or xo = 1, then, for k < n and 0 < |a; — xo| < m, < M[l + {e1'2

+ \x-xo\)a-k

+ e(e1'2

+ \x-xo\)-k-2]

, (3.67)

with 0 < (3 < c(xo)/|o'(®o)| a n ( i some m > 0. While if a(xo) = 0 and 0 < — pa'(xo) < c(zo), for p > 0 and xo = 0 or XQ = 1 then, analogously to (3.63), ( x , e ) | < M[ 1 + e where d = c(xo) +

! exp(-v/d|x-x0|/e1/'2)] ,

< TU , (3.68)

k 2

a'(xo).

Similarly, in the vicinity of an interior turning point xo we have < M ,

\x - xQ\

0 then | u W ( x , e ) | < M { 1 + [ e 1 / 2 + (x -

,

\x - x0\ < m ,

k c > 0, then the estimates in the vicinity of the turning points can be improved. For example, an estimate more accurate than (3.69) in the vicinity of an isolated interior turning point xq with a'(xo) < 0 in (3.71) was obtained by Berger, Han, and Kellogg (1984): l + (e 1 / 2 + | ® - a ; o | ) a - * , ik

U \x,£)I

< M
0,

¡s _ i 0 | ) | ( e i / 2 + ^ _

1,2,..., XQ\)a-k

)

(3.72) with a = c(xo)/|a'(a;o)| and \x — XQ\ < m for some m > 0. The estimate (3.72) was proved by employing appropriate cylinder mappings and a Green's function. This approach is suitable for an analysis of linear problems only. Note t h a t the technique of barrier finctions, considered in the book, is suitable for analyzing arbitrary nonlinear equations. Also more accurate than (3.67) there can be obtained an estimate of the solution derivatives to the Dirichlet problem for (3.71) in the vicinity of the boundary turning point x0 = 0 or xo = 1 with the condition a'(xo) < 0. In this case Liseikin (1984) has proved the following estimate l« ( f c ) (*»s)|


0, arbitrary m > 0, and 0 < a < c(xo)/|a'(a;o)|. Note that in the case of the linear equation (3.71) and the condition a(x) 0 for x € [0,1] the restriction c(x) > 0 is redundant. Since by the change u —> u e x p ^ the equation obtained from (3.71) with respect to v(x) observes this restriction, if an appropriate constant b is chosen.

3.3.2.

Problem with the Condition

f(x,u) =

xg(x,u)

Here we consider the following two-point boundary value problem L[u] = —£u" - xa(x)u'

+ xg(x,

u) = 0 ,

0 < a; < 1 ,

r[tt] = [ « ( 0 ) e ) , « ( l 1 e ) ] = ( A O ) A i ) ,

(3.74)

where a(x) € Cn[0,1]

,

g(x,u)

e C n ' " ( [ 0 , l ] x R) ,

l > e > 0 .

For this problem the condition (3.24) of strong ellipticity is not held.

Theorem 3.3.2.

Let u(x,e)

and gu{x,

Then

u) > 0.

be a solution

e)| < M{ 1 + £-k'2

0< x < 1 , for arbitrary

to

(3.74)

with a(x)

e x p { - m x / e 1 ' 2 ) + ( e 1 / 2 + x)l~k]

(3.75)

m > 0. If a(x) = 0 and gu{x, M{l

+

u) > c > 0,

then

£-k/3exp(-mx/£^3)

+ £ ~ k / 2 exp[s(a; - 1 ) / £ 1 / 2 ] + ( e 1 / 3 + x)2~k} 0 < x < 1 ,

Proof. Let

0 0, gu(x,u)

> 0. Using as a barrier the function

|A0| + |Ai| + (2 - x)

max 0 a

+xa(x)

m a x \g{x, 0 ) / a | > 0 . 0 L[-b]

,

0
r [ - 6 ] , and consequently 0 < a: < 1 ,

\u{x,e)\ 0 .

we obtain, in analogy with

fr[e,tt(e,e)]exp[p(g,s,g)]dc

where

(3.78)

(3.79)

X

X,e) = -£~l

J 7?a(i?)d77. i

From (3.74), (3.77), (3.78), and (3.79) we readily find that \u{k){x,e)\
a > 0