tumabstract the goal of this lecture is to introduce some fundamental notions and techniques used in...

Lecture notes, Winter term 2018/19

Fallstudien der Mathematischen Modellbildung:

Asymptotic methodsfor perturbation problems

Approximation techniques in science and engineering

Stefan Possanner ([email protected] )

February 5, 2019

Technical University Munich

Department of Mathematics

mailto:[email protected]

Abstract

The goal of this lecture is to introduce some fundamental notions and techniques usedin the asymptotic analysis of perturbation problems. Such problems are called singularif they undergo a change in their mathematical structure as the perturbation parame-ter ε tends to zero. A solution of the reduced problem (ε = 0) coincides with the limitsolution of the full problem as ε→ 0 only if the perturbation is regular. It is the subjectof asymptotic analysis to find approximate solutions of the full problem that are validuniformly for 0 < ε ≤ ε0, even if the perturbation is singular. Singular perturbationproblems usually arise at the most critical (and interesting) regimes of physical modeling- their analysis and ultimate resolution has often lead to major advances in a specificfield of science. In the first part of this course we focus on some basic principles andexamples in the context of ordinary differential equations: we introduce the principle ofdominant balance and discuss boundary layers, the WKB method, the method of (vari-ational) averaging and the method of multiple scales. The guiding-center approximationof plasma physics is considered as a generic example of nonlinear perturbation theory.In the second part we extend our analysis to partial differential equations and presentPrandtl’s boundary layer for the Navier-Stokes equation. Moreover, we elaborate onmacroscopic limits of kinetic equations in the strongly collisional regime, leading to fluidmodels of reduced dimensionality.

Contents

1. Introduction 51.1. Regular and singular problems: algebraic examples . . . . . . . . . . . . 61.2. The process of “non-dimensionalization” . . . . . . . . . . . . . . . . . . 91.3. General problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2. Asymptotic expansions 152.1. Order functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2. Order of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3. Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3. Regular perturbations 263.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2. Nonlinear initial-value problems . . . . . . . . . . . . . . . . . . . . . . . 263.3. Example: The nonlinear spring . . . . . . . . . . . . . . . . . . . . . . . 323.4. The method of the strained coordinate . . . . . . . . . . . . . . . . . . . 333.5. Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.1. Generic example . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5.2. K.B.M. Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5.3. The standard form . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6. The method of multiple scales . . . . . . . . . . . . . . . . . . . . . . . . 41

4. Singular perturbations of linear ODEs 424.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2. The initial value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3. The boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . 49

5. Macroscopic limits of kinetic equations 505.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2. The free transport equation . . . . . . . . . . . . . . . . . . . . . . . . . 515.3. Properties of the collision operator . . . . . . . . . . . . . . . . . . . . . 525.4. Moment equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A. Fundamentals of initial-value problems 57A.1. Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.2. Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.3. Transition map and local flow . . . . . . . . . . . . . . . . . . . . . . . . 59A.4. Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3

Contents

Index 63

4

1. Introduction

A mathematical problem is deemed a perturbation problem if it is “close” to a simplerproblem for which the solution is either known or can be computed with standard tech-niques. The closeness is usually measured in terms of a dimensionless parameter ε 1in the governing equations, which are typically systems of algebraic and/or differentialequations with suitable initial and boundary conditions, in a way that setting ε = 0yields the standard problem - henceforth called the reduced problem. In asymptoticanalysis, the approach is to view the solutions of the governing equations as functionsof ε, i.e. as a family of solutions depending on the parameter ε, and to construct anapproximation to this family in the form of a series expansion in terms of simple func-tions of ε (typically power series in εn, n ≥ 0). The big advantage is that this series canbe computed term by term from simplified equations and is thus much easier to obtainthan the exact solution. It occurs quite often that such an asymptotic expansion (AE)is the basis for a numerical investigation of the problem, which would otherwise be stiffor simply too large (in terms of degrees of freedom) to solve.

In regular perturbation problems the lowest order of the AE is indeed the solutionof the reduced problem with ε = 0 across the whole domain of interest. In this case,it is straightforward to derive a system of equations with suitable initial and boundaryconditions for the terms in the series, which can then be solved recursively. The lower thevalue of ε, the better the approximation obtained via the series expansion. We shall studyregular perturbations of nonlinear ordinary differential equations (ODEs) in Chapter 3.A perturbation problem that is not regular is called singular . For singular problems thelimiting behavior ε→ 0 is not captured by naive AE and the above procedure fails. InODEs for example, singular problems occur when the derivative of the highest order isof size ε or smaller, which leads to the formation of boundary layers in certain regionsof the domain as ε → 0. This is because the order of the reduced problem is less thanthe number of initial/boundary conditions. Hence, a more subtle treatment is requiredto capture the correct asymptotics uniformly in the domain of interest, which is thesubject of Chapter 4. Regular expansions may also fail when the domain is infinite,i.e. when small errors accumulate and become large over long times due to so-calledsecular terms . The method of averaging and the method of multiple scales can dealwith secularities and will be discussed in Chapter ??. With regard to partial differentialequations (PDEs), singular problems occur when the type of the PDE changes in thereduced problem, or when the boundary conditions are such that the reduced problemis ill-posed (has no unique solution). We will present some generic examples of linearelliptic and hyperbolic equations in Chapter ??. In particular, we shall revisit Prandtl’sanalysis (from 1904) of boundary layer formation around a body in a nearly inviscid

5

CHAPTER 1. INTRODUCTION

flow. This is a prime example of the power of asymptotic analysis for advancing hardphysical problems, where straightforward reasoning might stall. Finally, in Chapter 5we illustrate more examples in the context of kinetic-fluid transitions in gas dynamicsand magnetized plasmas.

1.1. Regular and singular problems: algebraic examples

The difference between regular and singular perturbation problems can be readily un-derstood by means of the following two algebraic examples.

Example 1. Consider the cubic equation

x3ε − xε + ε = 0 . (1.1)

We view the roots xε as a family of solutions depending on the parameter ε. The reducedproblem is obtained by setting ε = 0,

x30 − x0 = 0 , (1.2)

which yields the three roots x0 ∈ 0,±1. In order to produce an AE of the family xεwe assume

xε = x0 + ε x1 + ε2 x2 +O(ε3) . (1.3)

We will clarify the meaning of the symbol O(ε3) later, here it is sufficient to know thatit describes terms that tend to zero at least as ε3 when ε → 0 (we say these terms are“of order 3”). Inserting the AE into (1.1) and ordering terms in powers of ε yields

0 = (x0 + ε x1 + ε2 x2 + . . .)3 − x0 − ε x1 − ε2 x2 + ε+O(ε3)

= (x30 − x0) + ε (3x2

0x1 − x1 + 1) + ε2(3x20x2 + 3x0x

21 − x2) +O(ε3) .

(1.4)

Since we assume the expansion (1.3) to be valid for a finite interval (0, ε0] of ε-values,and moreover the functions εn are linearly independent, the coefficients in (1.4) mustvanish. This leads to the system

x30 − x0 = 0 ,

3x20x1 − x1 + 1 = 0 ,

3x20x2 + 3x0x

21 − x2 = 0 ,

...

In the first equation we recognize the reduced problem (1.2). The other equations arelinear for x1, x2, etc. and can be solved recursively:

x1 =1

1− 3x20

, x2 =3x0x

21

1− 3x20

, . . . .

6


The AEs for the three roots of (1.1) thus read

xI,ε = ε+O(ε3) ,

xII,ε = 1− ε

2− 3 ε2

8+O(ε3) ,

xIII,ε = −1− ε

2+

3 ε2

8+O(ε3) .

We leave it up to the reader to verify that these are indeed the first terms of the Taylorexpansions of the exact roots.

Example 2. Consider now the following cubic equation:

ε x3ε − xε + 1 = 0 . (1.5)

Inserting the ansatz (1.3) and equating the coefficients of εn to zero yields

−x0 + 1 = 0 ,

x30 − x1 = 0 ,

3x20x1 − x2 = 0 ,

...

such that the AE readsxε = 1 + ε+ 3 ε2 +O(ε3) .

Here, we obtained only one root of the cubic equation, because the problem degener-ates to a linear equation when setting ε = 0. Such a qualitative change in themathematical nature is typical for a singular perturbation problem. Whathappened to the other two roots? In the present case they cannot be described withan AE of the form (1.3) because they tend to infinity as ε → 0. In order to recoverthe correct asymptotics of these roots we need to reformulate the problem (1.5) into aregular perturbation problem. This can be done via a change of variables of the form

xε =yεδ(ε)

,

where yε = O(1) as ε→ 0 and the function δ(ε) is still to be determined. The equationfor yε reads

ε

δ(ε)3y3ε −

1

δ(ε)yε + 1 = 0 . (1.6)

To obtain a non-trivial reduced problem we require at least two leading-order terms in(1.6) to be of the same order in ε. This is called the principle of dominant balance,which we will encounter repeatedly in this course. Following this principle, there will be

7


one choice of δ(ε) which leads to meaningful results. Balancing the first two terms leadsto

ε

δ(ε)3=

1

δ(ε)=⇒ δ(ε) =

√ε .

With this choice of δ(ε) we obtain the regular perturbation problem

y3ε − yε +

√ε = 0 ,

which we can solve in the same way as (1.1) with the ansatz

yε = y0 +√ε y1 + ε y2 +O(ε3/2) ,

The non-zero solutions read

yε = ±1−√ε

2∓ 3 ε

8+O(ε3/2) ,

which yields the missing roots

xε = ± 1√ε− 1

2∓ 3√ε

8+O(ε) .

The limit ε→ 0 in such an AE with xε = O(ε−1/2) is called a distinguished limit .There are two other possibilities for balancing two terms in (1.5). Balancing the last

two terms leads to δ(ε) = 1 and thus to the original problem - this is clearly a badchoice. On the other hand, balancing the first and the third term leads to δ(ε) = ε1/3.However, in this case the second term would be of order O(ε−1/3) and thus dominatethe other two terms, which violates the principle of dominant balance. In this example,no three-term dominant balance is possible but this can occur in other problems.

Example 3. As a last algebraic example consider the quadratic equation

(1− ε)x2ε − 2xε + 1 = 0 .

Trying the ansatzxε = x0 + ε x1 +O(ε2) (1.7)

leads to

x20 − 2x0 + 1 = 0 ,

2x0x1 − x20 − 2x1 = 0 .

From the first equation we obtain x0 = 1 as a double root and then the second equationyields the contradiction −1 = 0. Hence, a solution of the form (1.7) cannot exist. Thedifficulty arises because x0 = 1 is a repeated root of the reduced problem and thus theexact solution

xε =1±√ε

1− ε

8


does not have a power series expansion in ε, but rather in√ε. An expansion of the form

xε = x0 +√ε x1 + ε x2 +O(ε3/2)

leads to x0 = 1 and x21 = 1, which yields the correct expansion

xε = 1±√ε+O(ε) .

1.2. The process of “non-dimensionalization”

The dimension-less parameter ε is a key factor in perturbation problems, as it allows togive a mathematical meaning to the transition between the reduced and the full prob-lem. However, science problems are usually written in terms of dimensional quantities,i.e. variables with physical dimensions such as time, length or temperature. Most prob-lems consist of equations that model real-world phenomena, and are thus linked at somepoint to experimental observations in which the model is rooted - hence the dimensionalnature of the variables. The first task in the perturbative treatment of a science problemis thus to write it in non-dimensional form, thereby identifying the parameter ε. Thiscrucial step enables the formulation of the “physical” problem as a well-defined per-turbation problem. Depending on the considered problem parameters, the same modelequations may lead to either a regular perturbation problem, a singular perturbationproblem, or not a perturbation problem at all (ε ≈ 1)! It is the physical information ofthe considered scenario, i.e. the size of the problem parameters, that in the end definesthe nature of the perturbation problem.

As we will see later, the term “asymptotic” means “for ε sufficiently small”. Hence, inmathematical terms, an asymptotic approximation - that is is an AE which is asymptot-ically equivalent to the exact solution (clarified later) - can be made arbitrarily preciseby rendering ε as small as necessary. By contrast, in a given non-dimensional scienceproblem the size of ε is fixed, usually denoting the ratio of two characteristic problemparameters. The validity of the asymptotic approximation then has to be checked forthis particular value of ε. This can be done for instance if explicit expression for theremainder of the series expansion are available.

Example 4. We will now familiarize ourselves with the concept of non-dimensionalization(or scaling) by considering the damped harmonic oscillator. Let x(t) denote the displace-ment of a mass m > 0 attached to spring from its equilibrium position as a function oftime. If the mass is set into motion from its equilibrium position with an impulse p0,the ensuing dynamics can be described in terms of the following initial value problem(IVP):

md2x

dt2+ β

dx

dt+ k x = 0 ,

x(0) = 0 ,dx

dt(0) =

p0

m.

(1.8)

9


Here, k > 0 stands for the spring constant and β ≥ 0 denotes the damping coefficient.Scaling the problem (1.8) starts with writing the dependent variable x and the indepen-dent variable t in terms of characteristic units of length L and of time T , respectively,

x(t) = Lx′(t′) , t = T t′ .

Here, x′ and t′ are dimension-less. From the chain rule we have

dx(t)

dt=L

T

dx′(t′)

dt′,

d2x(t)

dt2=

L

T 2

d2x′(t′)

(dt′)2.

Inserting this into (1.8) yieldsmL

T 2

d2x′

(dt′)2+βL

T

dx′

dt′+ kLx′ = 0 ,

L x′(0) = 0 ,L

T

dx′

dt′(0) =

p0

m.

Now this problem is still dimensional but the dimensions (or units) have been madeexplicit via the pre-factors of each term. The dimension-less form is obtained by dividingby one of the pre-factors which leads for example to

m

kT 2

d2x′

(dt′)2+

β

kT

dx′

dt′+ x′ = 0 ,

x′(0) = 0 ,dx′

dt′(0) =

p0T

mL.

Our next task is to identify the small parameter ε. Hence we need to assign, a priori, a“size” to each of the terms in the problem. This requires additional information regardingthe physical scenario we aim to consider. First of all, we identify two characteristic timescales of the problem:

τ1 =

√m

kand τ2 =

β

k.

Here, τ1 is the period of the undamped oscillator and τ2 is a characteristic dampingtime. With the choice of the time scale T we can determine which phenomena we wantto resolve on a scale of order one, thus T is also called the time scale of observation.Setting T = τ1 will resolve the oscillatory phenomena, while setting T = τ2 will lead usto observe the damping of the oscillator on a scale of order one. The length scale L canbe determined by requiring an initial velocity dx′/dt′ of order one, hence L = p0T/m.Let us compare two physical scenarios:

1. Weakly damped oscillator: τ2 τ1 and T = τ1. Setting

ε :=τ2

τ1

,

10


and omitting the primes for lighter notation, this leads to

P (osc)ε

d2xεdt2

+ εdxεdt

+ xε = 0 ,

xε(0) = 0 ,dxεdt

(0) = 1 .

(1.9)

As we will learn later this problem can be classified a regular perturbation problemas long as t is bounded, t ∈ [0, T ) with T independent of ε (more precisely, T =O(1) as ε→ 0). Indeed, trying the AE

xε = x0 + ε x1 +O(ε2)

leads to

P(osc)0

d2x0

dt2+ x0 = 0 ,

x0(0) = 0 ,dx0

dt(0) = 1 ,

P(osc)1

d2x1

dt2+ x1 = −dx0

dt,

x1(0) = 0 ,dx1

dt(0) = 0 .

The reduced problem yields x0(t) = sin(t) and then the problem P(osc)1 yields the

correction x1(t) = −t sin(t)/2. Therefore, the AE reads

xε = sin(t)− εt

2sin(t) +O(ε2) .

This is a valid expansion as long as εt is small compared to one, hence for finitetime t. Terms that blow up as t → ∞ are called secular terms in perturbationtheory. There are more sophisticated methods for (nearly-) periodic problemswhich can avoid secular terms in AEs (averaging, method of multiple scales).

2. Strongly damped oscillator: τ1 τ2 and T = τ2. We define

ε :=

(τ1

τ2

)2

,

which leads to

P (dmp)ε

ε

d2xεdt2

+dxεdt

+ xε = 0 ,

xε(0) = 0 ,dxεdt

(0) = 1 .

(1.10)

This is a singular perturbation problem because the highest order derivative is multipliedby ε. The reduced problem reads

P(dmp)0

dx0

dt+ x0 = 0 ,

x0(0) = 0 ,dx0

dt(0) = 1 .

11


Figure 1.1.: Exact solutions for the weakly damped oscillator (1.9) in (a) and for thestrongly damped oscillator (1.10) in (b), for different values of ε.

This problem is obviously ill-posed because x0 cannot satisfy both initial conditions.Such a situation, where the order of the differential equation is decreased in the reducedproblem, leads to the formation of boundary layers . The terminology becomes clear bylooking at the exact solutions of P

(dmp)ε for different values of ε in panel (b) of Figure 1.1.

Boundary layer problems can be treated with asymptotic matching, cf. Chapter 4.

1.3. General problem setting

In this course we are faced with finding approximate solutions to mathematical problemsas a small parameter ε > 0 tends to zero. We write these problems symbolically as

Pε[uε] = 0 . (1.11)

Here, uε is the solution of the problem and Pε represents a set of model equations. Inthis course we focus on ordinary and partial differential equations (ODEs and PDEs).Hence, uε(x) is defined for x ∈ D ⊂ Rn and we write uε(x) = u(x1, . . . , xn, ε); Pε issome differential operator with suitable initial/boundary conditions. Often times, theproblem (1.11) is too hard to solve, even with the help of numerical methods. The goalof asymptotic analysis is to find ”simple” approximations for uε when ε is small. Theterm simple is subjective here; for instance, it could mean approximating the solutionin terms of elementary functions or finding an approximation that can be computed ona lower dimensional subset D ⊂ D. Setting formally ε = 0 leads to the reduced problem

P0[u0] = 0 .

An immediate question arises:

12


Is the limit of the solution equal to the solution of the limit?

As already hinted before, there are two cases:

regular problem: ∀ x ∈ D limε→0

uε(x) = u0(x) ,

singular problem: ∃ x0 ∈ D s.t. limε→0

uε(x0) 6= u0(x0) .

From this it is clear that a naive expansion of the form

uε = u0 + εu1 + ε2u2 + . . . (1.12)

cannot be uniformly valid in D when a problem is singular. In the latter case, a viablestrategy is to reformulate the equations into a regular perturbation problem and thenapply an expansion of the form (1.12) to approximate the solution. This is commonly thestrategy for developing asymptotic-preserving (AP) numerical schemes for stiff equations[8, 5]. If such a reformulation is not available, one distinguishes two types of problems:singular problems of cumulative type and singular problems of boundary layer type.

Singular problems of cumulative type. Theses are problems with oscillatingsolutions where the influence of the small parameter ε on the limit solution becomesobservable only after long times of the order t = O(1

ε). The error terms are called

secular terms and blow up as time goes to infinity, but are small for times of order one.The domain D is infinite, for instance D = R for ODEs. The secular terms lead to thefollowing behavior:

limε→0

uε(t) 6= u0(t) for t = O(1

ε

).

The techniques we will discuss in this course for dealing with problems of cumulativetype are

• classical averaging,

• variational averaging for Lagrangian dynamical systems,

• the method of multiple scales.

Most of these techniques have been developed for studying the motion of celestial bodiesand date back to the times of Poincare (∼ 1900). The technique of variational averaging[1, 10] has gained renewed attention for studying the helical motion of a charged particlein a strong magnetic field, a classical example for a problem of cumulative type.

Singular problems of boundary layer type. Many interesting phenomena inphysics are characterized by a sudden change of state variables, for instance the formationof shock waves in gas dynamics or the boundary layer flow along the surface of a body.Mathematically, such problems can be described as singular perturbation problems wherethe domain D is finite. As ε tends to zero, the solution develops a jump in a very narrow

13


region of D, called the boundary layer. The boundary layer can be located at the edgesof the domain but it does not have to be (free boundary layer problem). The main toolfor treating such problems is called

• asymptotic matching.

14

2. Asymptotic expansions

Let us now introduce the main tools for asymptotic analysis. We shall give a precise def-inition of the terms appearing in expansions of the form (1.12) and generalized versionsthereof. Moreover, we must clarify what is meant by an asymptotic expansion (AE) ofuε. An interesting concept will be that of an asymptotic series, which in general doesnot converge, but nevertheless provides a good approximation to uε for small ε. Thedifferent notions of convergence and asymptotic convergence will be compared in detail.

2.1. Order functions

Definition 1. Let E be the set of real functions δ(ε) that are strictly positive andcontinuous on the interval (0, ε0] and such that

1. limε→0 δ(ε) exists (it can be ∞),

2. δ1, δ2 ∈ E ⇒ δ1δ2 ∈ E.

A function δ(ε) ∈ E is called an order function.

The following functions are examples of order functions:

1, ε, 1 + ε, ε3,1

ε,

ε

1 + ε,

1

ln(1/ε), e−1/ε .

Note that if δ(ε) is an order function, then 1/δ(ε) is too. The first condition above accepts1/ε, but it excludes functions with rapid variations near zero such as 1 + sin2(1/ε). Thesecond condition excludes products of such functions with viable order functions, likeε[1 + sin2(1/ε)]. A comparison of order functions is accomplished via Hardy’s notation:

δ1 is asymptotically smaller than δ2, δ1 ≺ δ2, if limε→0

δ1

δ2

= 0 ,

δ1 is asymptotically equal to δ2, δ1 ∼ δ2, if limε→0

δ1

δ2

= λ, 0 < λ <∞ ,

δ1 is asymptotically smaller than or equal to δ2, δ1 δ2, if limε→0

δ1

δ2

= λ, 0 ≤ λ <∞ .

Example 5. Using Hardy’s notation we have, for n ∈ N0:

εn+1 ≺ εn, e−1/ε ≺ εn ≺ 1

ln(1/ε),

2εn ∼ εn, 2ε ∼ ε

1 + ε, 2 ∼ 1 + ε, ε ∼ sin(ε) .

15

CHAPTER 2. ASYMPTOTIC EXPANSIONS

Definition 2. A sequence of order functions δn is called asymptotic sequence if

δn+1 ≺ δn ∀n .

If δn and γn are two asymptotic sequences with δn ∼ γn for all n, than these sequencesare asymptotically equivalent. For example,

εn,

(ε

1 + ε

)n, sinn(ε) ,

are asymptotically equivalent. The relation δ1 ∼ δ2 defines an equivalence relation on E,meaning it is reflexive, symmetric and transitive. Hence we can choose representativesof each class of equivalence. Usually we will work with the most convenient set oforder functions like εn where n is an integer or εα where α is rational, and call theserepresentatives gauge functions . The choice of gauge functions has implications on theuniqueness of asymptotic expansions discussed below.

2.2. Order of a function

Let us consider real-valued, continuous functions u(ε) for ε ∈ (0, ε0]. The magnitudeof these functions as ε → 0 can be compared to order functions δ ∈ E by means ofLandau’s “big-Oh” and “small-oh” notation:

Definition 3. (Order of ε-functions.)

(i) u = O(δ) as ε→ 0 if there exist positive constants k and C such that |u(ε)| ≤ kδ(ε)for 0 < ε < C. (Remark that the limit limε→0 |u(ε)|/δ(ε) need not exist).

(ii) u = o(δ) as ε→ 0 if limε→0 u(ε)/δ(ε) = 0.

(iii) u = Os(δ) if u = O(δ) and u 6= o(δ) at x = x0.

If u is an order function then Hardy’s and Landau’s notation are equivalent. Landau’snotation is however more general, because the limit in (i) need not exist, for instance wehave sin(1/ε) = O(1) in Landau’s notation.

In what follows we consider real-valued functions u(x, ε) = uε(x) where x ∈ D ⊂ Rn

and ε ∈ (0, ε0]. For ε fixed, we suppose that uε : D → R belongs to a normed linearspace with the norm || · || and that ||uε|| is continuous in ε. We want to compare suchfunctions to order functions δ ∈ E as ε → 0. This can be done either point-wise, oruniformly in some subset D0 ⊂ D, or by using the norm || · ||.Definition 4. (Order, point-wise.) Let x0 ∈ D.

(i) u = O(δ) at x = x0 as ε → 0 if there exist positive constants k and C such that|u(x0, ε)| ≤ k|δ(ε)| for 0 < ε < C.

(ii) u = o(δ) at x = x0 as ε→ 0 if limε→0 u(x0, ε)/δ(ε) = 0.

(iii) u = Os(δ) at x = x0 if u = O(δ) and u 6= o(δ) at x = x0.

16


Definition 5. (Order, uniformly.) Let D0 ⊂ D.

(i) u = O(δ) uniformly in D0 as ε → 0 if there exist positive constants k and C,independent of x, such that for all x ∈ D0, |u(x, ε)| ≤ k|δ(ε)| for 0 < ε < C.

(ii) u = o(δ) uniformly in D0 as ε→ 0 if limε→0 u(x, ε)/δ(ε) = 0 for x ∈ D0.

(iii) u = Os(δ) uniformly in D0 if u = O(δ) and u 6= o(δ) uniformly in D0.

Definition 6. (Order, in norm.) Let D0 ⊂ D and suppose that the restriction of uεto D0 belongs to a normed linear space with norm ||uε||D0 .

(i) u = O(δ) in D0 as ε→ 0 if ||uε||D0 = O(δ).

(ii) u = o(δ) in D0 as ε→ 0 if ||uε||D0 = o(δ).

(iii) u = Os(δ) in D0 as ε→ 0 if u = O(δ) and u 6= o(δ) in D0.

In the last definition the order of a function depends on the chosen norm || · ||D0 , whichshould be naturally related to the norm || · || for functions on D. The supremum normis used most frequently in asymptotic analysis, that is, for uε continuous and boundedin D and D0 ⊂ D,

||uε||D0 = maxx∈D0

|uε(x)| .

However, other norms such as L2 can be used, depending on the type of problem one isinterested in. The order of a function can then be completely different, depending onthe norm chosen. Consider for example u(x, ε) = uε(x) = e−x/ε on D0 = [0, 1]. In thesupremum norm we have u = O(1), whereas in the L2-norm,

||uε||D0 =

(∫ 1

0

u2ε(x) dx

)1/2

,

we have u = O(√ε). In what follows we shall always use the supremum norm if

not stated otherwise.

Remark 1. It is clear that Definition 5 is a special case of Defintion 6 with || · ||D0 thesupremum norm.

In the analysis of singular perturbation problems the order of a function is usually notuniform in the whole domain D. This is why the notion of order in a subset D0 ⊂ D isimportant and has been stressed in the above defintions. This will become particularlyapparent during the analysis of boundary layers. We close this section with an importantLemma:

Lemma 1. Suppose u(x, ε) = uε(x) such that uε : D → R belongs to a normed linearspace and that ||uε|| is continuous in ε ∈ (0, ε0]. Then there exists an order functionδ ∈ E such that u = Os(δ) in D.

17


Proof. See for example Eckhaus [7].

Under the conditions of Lemma 1 we can always rescale the function u such that

u

δ= Os(1) . (2.1)

2.3. Asymptotic expansions

We will now clarify the notions of

• asymptotic approximation (AA) ,

• (finite) asymptotic series ,

• asymptotic expansion (AE).

These terms are often used synonymously in the literature, however some clarificationis provided for example by Eckhaus [7].

Definition 7. Let u(x, ε) be a function that is Os(1) in D. A function u0(x, ε) is anasymptotic approximation (AA) of u if

u− u0 = o(1) in D .

Moreover, u0 is called a regular AA if it is independent of ε, i.e. u0 = u0(x).

Remark that if a regular AA of u exists it is unique, given by u0(x) = limε→0 u(x, ε).AAs can be defined for functions u that are of arbitrary sharp order δ via the rescalingprocedure (2.1). In general, u0 is an asymptotic approximation of u = Os(δ) in D if

u− u0

δ= o(1) in D .

This implies u0 = Os(δ).

Example 6.

• u0(x, ε) = 1 is a regular AA of u(x, ε) = eεx on any bounded interval [−A,A] withA independent of ε, because

limε→0

(u− u0) = limε→0

(eεx − 1) = limε→0

(1 + εx+1

2ε2x2 +O(ε3)− 1) = 0 in [−A,A] .

• u0(x, ε) = εx + ε2 cos(3x) is an AA of u(x, ε) = sin(εx) on any bounded interval[−A,A] with A independent of ε, because

limε→0

u− u0

ε= lim

ε→0

sin(εx)− εx− ε2 cos(3x)

ε

= limε→0

εx− 16ε3x3 +O(ε5)− εx− ε2 cos(3x)

ε

= limε→0

(−1

6ε2x3 − ε cos(3x)) = 0 in [−A,A] .

18


An asymptotic expansion of u is constructed by successive AAs of the remainderterms. For example, let u0 = Os(1) be an AA of u in the sense of Definition 7. Then wedefine the remainder as

R1 := u− u0 = o(1) .

It follows from Lemma 1 that there exists a δ1 ≺ 1 such that R1 = Os(δ1), and we canrescale via (2.1) to obtain R1 := R1/δ1 = Os(1). Therefore, we can write

u = u0 + δ1R1 . (2.2)

Suppose now we are able to find an AA of the remainder R1(x, ε) and call it u1(x, ε).Then from Definition 7 we define the new remainder as

R2 := R1 − u1 = o(1) ,

and there exists a δ2 such that R2 = Os(δ2). Rescaling leads to R2 := R2/δ2 = Os(1)and by inserting this into (2.2) we can write

u = u0 + δ1u1 + δ1δ2R2 .

We can repeat this procedure N times to obtain

u(x, ε) =N∑n=0

δn(ε)un(x, ε) + o(δN(ε)) , (2.3)

where δ0 = 1, δn≥1(ε) = δ1(ε) · . . . · δn(ε) and un(x, ε) = Os(1). Remark that the productof order functions is again an order function due to the property 2 of Definition 1. Theright-hand side in (2.3) is called an asymptotic expansion (AE) of the function u in D.It follows in particular that

limε→0

||u(x, ε)−∑N

n=0 δn(ε)un(x, ε)||δN(ε)

= 0 .

Let us now formalize this result:

Definition 8. Let (u∗n(x, ε))∞n=0 denote a sequence of functions on D × (0, ε0] withu∗n = Os(δn). A (finite) series u(N) given by

u(N)(x, ε) =N∑n=0

u∗n(x, ε) , u∗n = Os(δn) in D ,

is called an asymptotic series in D if δn is an asymptotic sequence, hence if δn+1 ≺ δn(see Definition 2).

By rescaling as in (2.1) one obtains u∗n = δnun with un = Os(1), hence any asymptoticseries can be written as

u(N)(x, ε) =N∑n=0

δn(ε)un(x, ε) .

19


Example 7. The expressions

f (2)(x, ε) = εx− ε2x2

2+ε3x3

3,

g(2)(x, ε) = 1 +ε2x2

2+

ε4

(1 + ε)4

x4

24,

h(2)(x, ε) = 1 + εx+ sin(ε2x2) ,

are finite asymptotic series in any bounded interval [−A,A] with A independent of ε.

Definition 9. Let u(N) denote a (finite) asymptotic series in D given by

u(N)(x, ε) =N∑n=0

δn(ε)un(x, ε) , un = Os(1) in D ,

then u(N) is an asymptotic expansion (AE) to order δN of u in D if

u− u(N) = o(δN) in D .

Moreover, u(N) is called a regular AE if it is of the form

u(N)(x, ε) =N∑n=0

δn(ε)un(x) ,

hence if the coefficient functions un do not depend on ε.

Example 8. For the three asymptotic series given in Example 7 we have

ln(1 + εx)− f (2)(x, ε) = o(ε3) in [−A,A] ,

cosh(εx)− g(2)(x, ε) = o(ε4) in [−A,A] ,

1

1− εx− h(2)(x, ε) = o(ε2) in [−A,A] .

Hence, f (2) is an AE to order ε3 of ln(1 + εx) in [−A,A], g(2) is an AE to order ε4 ofcosh(εx) in [−A,A] and h(2) is an AE to order ε2 of 1/(1 + εx) in [−A,A]. This followsimmediately from the series expansions

ln(1 + x′) =∞∑n=1

(−1)n+1 (x′)n

n|x′| < 1 ,

cosh(x′) =∞∑n=0

(x′)2n

(2n)!x′ ∈ R ,

1

1− x′=∞∑n=0

(x′)n |x′| < 1 .

Remark that the series expansions have a certain radius of convergence, whereas theAEs are valid in the whole interval [−A,A]. This is a first hint towards the differencebetween convergence and asymptotic convergence.

20


If the asymptotic expansion in Definition 9 holds for any positive integer N one writes

u ∼∞∑n=0

δn un ,

and says the series is asymptotically convergent to u in D. Be mindful however thatasymptotic convergence does not imply convergence of the series in the usualsense! Indeed, most asymptotic series do not converge, but this is also not their purpose,which is is to provide good approximations to a function as ε→ 0. Let us compare thetwo notions of convergence with respect to ε in detail. Consider x0 ∈ D to be fixed:

• Convergence of a series: the limit

limN→∞

N∑n=0

δn(ε)un(x0, ε)

exists for ε ∈ (0, R), where R is the radius of convergence.

• Asymptotic convergence to an ε-function u(x0, ε):

limε→0

∣∣∣u(x0, ε)−∑N

n=0 δn(ε)un(x0, ε)∣∣∣

δN(ε)= 0 ∀ N ∈ N .

We see that convergence is about the behavior of the series in a finite ε-region (0, R) asN →∞, whereas asymptotic convergence is about the approximation of the ε-functionu(x0, ε) as ε tends to zero. While the former is an absolute concept, the latter is arelative concept, always with respect to a given function u. It thus makes no sense to aska question like ’Is this series asymptotically convergent?’ A reasonable question wouldbe ’Is this series asymptotically convergent to u?’

As we have already seen in Example 8, the most common examples of AEs are Taylorexpansions. Suppose u is N -times differentiable at ε = 0, then Taylor’s theorem states

u =N∑n=0

εn1

n!

dnu

dεn

∣∣∣ε=0

+ o(εN) . (2.4)

In case that N becomes infinite the series can be convergent within a certain radiusR around zero and divergent elsewhere (R can be zero). However, the finite Taylorexpansion (2.4) is always a good approximation of the function u if one is sufficientlyclose to ε = 0.

Example 9. Given a function f : R→ R which is f = O(1) in R, we have

eεf(x) = 1 + εf(x) +ε2f 2(x)

2+ . . .+

εnfn(x)

n!+ . . . .

21


This series converges for all values of ε. The AE

eεf(x) = 1 + εf(x) +ε2f 2(x)

2+ o(ε2) in R ,

is valid near ε = 0 but not elsewhere.

Example 10. The order of a function u : D × (0, ε0]→ R might not be uniform in thewhole domain D. Consider for instance the function

u(x, ε) =√x+ ε =

√x ·√

1 +ε

x, x > 0 .

Taylor expansion leads to

u(x, ε) ∼√x

(1 +

ε

2x− ε2

8x2+ . . .+

(−1)n−1(2n− 3)!!

2nn!

εn

xn+ . . .

).

This is an AE of u in any left bounded interval x ∈ [A,∞) with A > 0; however it isnot an AE for x ∈ (0,∞) because the remainder terms are not o(εn) for x = O(ε).

Example 11. Consider the error function defined by

erf (t) = 1− 2√π

∫ ∞t

e−s2

ds .

It can be shown ([4], page 16) that this function can be approximated by

erf (t) = 1− e−t2

√π

N∑n=1

(−1)n−1 (2n− 3)!!

2n−1

1

t2n−1+ o( 1

t2N−1

),

which is true for any value of N ∈ N. Hence, substituting t = 1/ε, the following seriesis asymptotically convergent to the error function,

erf(1

ε

)∼ 1− e−1/ε2

√π

∞∑n=1

(−1)n−1 (2n− 3)!!

2n−1ε2n−1 .

This is an AE of erf with respect to the sequence (ε2n+1e−1/ε2)∞n=0 and it diverges for anyvalue of ε. It is nevertheless a useful AE for obtaining values of the error function forlarge arguments by keeping only the first few terms.

Example 12. The literature on special functions is full of useful AEs of these functions.Consider for instance the Bessel function J0(t), which has the series expansion

J0(t) =∞∑n=0

(−1)n( tn

2nn!

)2

.

22


This AE is convergent on any bounded interval of R. For large t, writing t = 1/ε, wehave the well-known AE ([4], page 17)

J0

(1

ε

)∼√

2ε

π

cos(1

ε− π

4

) ∞∑n=0

(−1)n(4n− 1)!!2

26n(2n)!ε2n

sin(1

ε− π

4

) ∞∑n=0

(−1)n(4n+ 1)!!2

26n+3(2n+ 1)!ε2n+1

.

While the convergent expansion is rather useless for for obtaining values of J0 for largearguments, the AE is very useful. To approximate the value of J0(3) to three digitsprecision one needs eight terms of the convergent series but only one term of the AE.

It is clear from the computations leading to (2.3) that an asymptotic expansion (AE)can always be studied as a repeated process of asymptotic approximations (AAs). Ifthese AAs are all regular and if we fix the δn to a certain set of gauge functions, forexample δn = εn, it follows that the regular AE of a function u, if it exists, isunique with respect to the chosen gauge functions, because the coefficientsun are uniquely determined by

un(x) = limε→0

u(x, ε)−∑n−1

m=0 δm(ε)um(x)

δn(ε).

The converse is however not true: a given asymptotic series u(N) is indeed the AE ofan infinity of functions which differ by a term of o(δN). These statements regardinguniqueness even hold for asymptotically convergent series. Suppose for example that wechoose δn = εn as gauge functions, then

u ∼∞∑n=0

εn un ⇔ u+ e−1/ε ∼∞∑n=0

εn un .

Two functions which have the same AE with respect to a given asymptotic sequence ofgauge functions are called asymptotically equal . Hence u is a.e. to u+e−1/ε with respectto the sequence εn. It may thus very well be that

u(x, ε) 6=∞∑n=0

δn(ε)un(x, ε) , x ∈ D, ε ∈ (0, R) ,

in case that the series converges. Finally, it is clear that the AE of a function changeswhen the asymptotic sequence δn changes.

In the special case that δn = εn and the AE is regular, one calls

u(x, ε) =N∑n=0

εnun(x) + o(εN)

the Poincare expansion (sometimes also Hilbert expansion) of u. Poincare expansionsare important tools in asymptotic analysis because of their relative simplicity; a lot can

23


be gained by approximating a complicated function of x and ε by a sequence of functionswith the simple structure εnun(x).

Another interesting question is the following: given a divergent AE that is asymp-totically convergent to u, what is the optimal truncation of the series for a fixed valueof ε? Optimal in this sense means with minimal error in the supremum norm. Hence wesearch for the optimal number Nopt after which to truncate the series to get the smallesterror. Such questions arise frequently in physics problems, where ε is usually defined bythe fraction of some fixed problem parameters.

Let us now consider some elementary operations on AEs like addition, multiplication,integration and differentiation. Let us assume expansions of Poincare type,

u(x, ε) ∼∞∑n=0

δn(ε)un(x) , vε(x) ∼∞∑n=0

δn(ε)vn(x) in D .

Then the following is true:

• Addition:

u(x, ε) + v(x, ε) ∼∞∑n=0

δn(ε)[un(x) + vn(x)] ,

• Multiplication: if δnδm = δn+m then

u(x, ε)v(x, ε) ∼∞∑n=0

δn(ε)wn(x) , wn(x) =n∑

m=0

um(x)vn−m(x) .

• Integration along a path γ in D: assuming everything is integrable along γ,∫γ

u(x, ε)dσ ∼∞∑n=0

δn(ε)

∫γ

un(x)dσ ,

where dσ is the line element along γ : I ⊂ R→ D.

• Integration with respect to ε: assuming everything is integrable w.r.t ε,∫ ε

0

uε′(x)dε′ ∼∞∑n=0

un(x)

∫ ε

0

δn(ε′)dε′ .

• Differentiation with respect to x: if u(x, ε) and un(x) are differentiable in D,

∂u(x, ε)

∂xi∼

∞∑n=0

δn(ε)∂un(x)

∂xi.

24


• Differentiation with respect to ε: if ∂u(x,ε)∂ε

for x ∈ D and dδn(ε)dε

exist for ε ∈ (0, ε0],and if

∂u(x, ε)

∂ε∼

∞∑n=0

dδn(ε)

dεu′n(x) ,

then u′n(x) = un(x) for x ∈ D.

The proof of these properties is straightforward.

25

3. Regular perturbations

3.1. Generalities

We consider initial/boundary-value problems of the form

Pε

Lε[uε(x)] = 0 , x = (x1, . . . , xn) ∈ D ⊂ Rn ,

Bε[uε(x)] = 0 , x ∈ S ⊂ ∂D ,(3.1)

where uε : D → Rm stands for the (vector-valued) solution of the problem, Lε representssome differential operator and Bε represents initial- and/or boundary conditions. Thedomain of the independent variables is D and the limiting conditions are prescribed ona subset S of the domain boundary ∂D. In the case of ODEs (n = 1), we distinguishinitial-value problems (IVPs), where the boundary conditions are prescribed at one pointx0 ∈ S, and boundary-value problems (BVPs), where the conditions are prescribed atmore than one point. Both Lε and Bε contain a small parameter ε and thus the problemPε is called a perturbation problem. The reduced problem is obtained by setting formallyε = 0 in Pε, leading to

P0

L0[u0(x)] = 0 , x ∈ D ⊂ Rn ,

B0[u0(x)] = 0 , x ∈ S ⊂ ∂D .(3.2)

The question we shall address is whether the solution u0 of (3.2) is a good approximationfor the solution uε of (3.1) in case that ε is small, and whether it is possible to improveon the approximation and to estimate the error. A first attempt to solve this problemis to assume a solution of the form

uε(x) =N∑n=0

εnun(x) +RN(x, ε) , (3.3)

and to try to solve for the coefficient functions u0,..., uN by substituting the ansatz (3.3)into Pε and sorting in powers of ε. If this succeeds we can try to estimate the error termRN . In case we can prove that RN = o(εN) in D, the solution (3.3) is a regular AE ofthe true solution and the perturbation problem Pε is called regular .

3.2. Nonlinear initial-value problems

Many perturbation problems related to research or engineering problems can be charac-terised as IVPs with a perturbed vector field (right-hand side). We mention for example

26

CHAPTER 3. REGULAR PERTURBATIONS

dynamical systems from physics, described by Newton’s equation of motion with a per-turbed force, or predator-prey models from biology. The abundance of such problemsin science is the reason we start with study of the regularly-perturbed IVPs as an entrypoint into the theory of asymptotic methods. As prerequisites we assume a basic under-standing of IVPs in order to quickly proceed with the asymptotic analysis. To refreshmemory, some basic material on IVPs has been gathered in appendix A.

In what follows the domain is D ⊂ R and the independent variable is denoted byt ∈ D. We search (vector-valued) functions uε : D → Rm that satisfy the IVP

Pε

duεdt

= f(uε, t, ε) ,

uε(t0) = V (ε) ∈ Rm , t0 ∈ D .

(3.4)

Here, V is the initial state of the variable uε and f : Rm × D × (0, ε0] → Rm is calledthe ”vector field” of the IVP. The vector field f is assumed to be

• N times continuously differentiable in Rm ×D × (0, ε0], with N ≥ 1.

• Lipschitz continuous with respect to the first argument in any bounded domainΩ ⊂ Rm,

||f(x, t, ε)− f(y, t, ε)|| ≤ Lf ||x− y|| x, y ∈ Ω . (3.5)

Here, || · || is some vector norm in Rm and Lf denotes the Lipschitz constant whichdoes not depend on (t, ε).

Moreover, we assume that f and V have the AEs

f(u, t, ε) =N∑n=0

εnfn(u, t) + o(εN) in Rm ×D ,

V (ε) =N∑n=0

εnVn + o(εN) .

(3.6)

It follows from Theorem 4 in Appendix A that under the above assumptions, the problemPε has a unique solution uε in some maximal interval Iε = Iε(V, t0) ⊂ D. For simplicity,let us restrict this solution to an interval t0 ≤ t ≤ t0 +Tε where we consider only forwardpropagation in time.

Our first instinct is that Pε is a regular perturbation problem, thus we try the regularansatz

uε(t) =N∑n=0

εnun(t) +RN(t, ε) , RN = o(εN) in D . (3.7)

27


Since the fn in (3.6) are CN -functions, we may apply Taylor’s theorem to compute

fn(uε, t) = fn

(u0 +

N∑n=1

εnun +RN , t

)

= fn(u0, t) +N∑k=1

1

k!

∂(k)fn∂xk

(u0, t) ·

(N∑n=1

εnun

)k

+ o(εN) .

(3.8)

Here, ∂fn/∂x ∈ Rm×m denotes the Jacobian matrix with respect to the first argumentof fn and the ∂(k)fn/∂x

k for k > 1 are multi-linear tensors of order k + 1 in the samesense. The dot · indicates the inner product of theses tensors with k vectors, yieldingagain a vector in Rm. The error term RN has been moved into the o(εN)-terms in thesecond line of (3.8). Substitution of (3.6)-(3.8) into the IVP (3.4) leads to

d

dt

(u0 + εu1 + ε2u2 + . . .

)=

= f0(u0, t) +∂f0

∂x(u0, t) ·

N∑n=1

εnun +1

2

∂2f0

∂u2(u0, t) ·

(N∑n=1

εnun

)2

+ . . .

+ ε

f1(u0, t) +∂f1

∂x(u0, t) ·

N∑n=1

εnun +1

2

∂2f1

∂u2(u0, t) ·

(N∑n=1

εnun

)2

+ . . .

+ ε2

f2(u0, t) +∂f2

∂x(u0, t) ·

N∑n=1

εnun +1

2

∂2f2

∂u2(u0, t) ·

(N∑n=1

εnun

)2

+ . . .

+ . . . ,

(3.9)

and(u0 + εu1 + ε2u2 + . . .)(t0) = V0 + εV1 + ε2V2 + . . .

for the initial conditions. Sorting (3.9) in powers of ε and equating to zero yields thefollowing equations:

du0

dt= f0(u0, t) ,

du1

dt=∂f0

∂x(u0, t) · u1 + f1(u0, t)︸︷︷︸

=:f1(u0,t)

,

du2

dt=∂f0

∂x(u0, t) · u2 +

1

2

∂2f0

∂u2(u0, t) · u2

1 +∂f1

∂x(u0, t) · u1 + f2(u0, t)︸︷︷︸

=:f2(u0,u1,t)

.

We see the evolving pattern and find the equation for the n-th order to be

dundt

=∂f0

∂x(u0, t) · un + fn(u0, . . . , un−1, t) . (3.10)

28


Here, fn is a function of depending on lower order solutions uk with k ≤ n− 1 only, andcan thus be viewed as a source term in the equation for un. This leads to the followingset of IVPs:

P0

du0

dt= f0(u0, t) ,

u0(t0) = V0 ,

(3.11)

P1

du1

dt=∂f0

∂x(u0, t) · u1 + f1(u0, t) ,

u1(t0) = V1 ,

...

Pn

dundt

=∂f0

∂x(u0, t) · un + fn(u0, . . . , un−1, t) ,

un(t0) = Vn ,

(3.12)

...

for n ≤ N . (3.11) is the reduced problem; due to our assumptions on f , which carryover to f0, the reduced problem has a unique solution in some interval t0 ≤ t ≤ t0 + T .The higher order problems Pn for 1 ≤ n ≤ N are all linear and are thus easily solved.Therefore, by means of the regular ansatz (3.7), the perturbation problem(3.4) has been transformed into a system of linear equations for the un, n ≥ 1,which can be solved sequentially once the solution u0 of the reduced problemis known. The solution of the linear problems Pn for n ≥ 1 can be written explicitlywith Duhamel’s formula. First, assuming fn = 0, the solution to the homogeneousproblem for any n reads

un(t) = exp

(∫ t

t0

∂f0

∂x(u0(s), s) ds

)· un(t0) .

Substituting the initial condition and defining the matrix

A(t, t0) := exp

(∫ t

t0

∂f0

∂x(u0(s), s) ds

)∈ Rm×m ,

we write the solution to the homogeneous problem as

un(t) = A(t, t0) · Vn .

Duhamel’s formula for the inhomogeneous problem Pn reads

un(t) = A(t, t0) · Vn +

∫ t

t0

A(t, s) · fn(u0(s), . . . , un−1(s), s) ds . (3.13)

29


Let us verify that this is indeed a solution of (3.12):

dundt

=dA(t, t0)

dt· Vn +

d

dt

∫ t

t0

A(t, s) · fn(u0(s), . . . , un−1(s), s) ds

=dA(t, t0)

dt· Vn +

∫ t

t0

dA(t, s)

dt· fn(u0(s), . . . , un−1(s), s) ds

+ A(t, t) · fn(u0, . . . , un−1, t)

=∂f0

∂x(u0(t), t)

[A(t, t0) · Vn +

∫ t

t0

A(t, s) · fn(u0(s), . . . , un−1(s), s) ds

]+ A(t, t) · fn(u0, . . . , un−1, t)

=∂f0

∂x(u0, t)un + fn(u0, . . . , un−1, t) .

We have thus computed the coefficient functions un of our ansatz (3.7) in formula (3.13).In order to show that (3.7) is indeed an AE of the true solution uε we now have to

prove that RN = o(εN). For this we aim to derive an IVP for the remainder RN andthen estimate how its size evolves using Gronwall’s lemma, stated below. First, let usdenote the solution we computed in terms of the un by

u(N)(t, ε) =N∑n=0

εnun(t) t0 ≤ t ≤ t0 + T ,

with u0 the solution of the reduced problem and the un for n ≥ 1 given in (3.13). Bygoing backwards through the steps in (3.10)-(3.8), we find that

du(N)

dt=

N∑n=0

εndundt

= f(u(N), t, ε) + o(εN) ,

where f is the vector field of the original IVP (3.4). Moreover, initially

u(N)(t0) =N∑n=0

εnVn . (3.14)

Therefore, from RN = uε − u(N), we obtain the following IVP for the remainder:dRN

dt= f(u(N) +RN , t, ε)− f(u(N), t, ε) ,

RN(t0) = o(εN) ,

for t0 ≤ t ≤ t0 + T . Hence, RN(t) ∈ Ω ⊂ Rm bounded during this time. Integrationwith respect to time yields

RN(t, ε) =

∫ t

t0

[f(u(N)(s, ε) +RN(s, ε), s, ε)− f(u(N)(s, ε), s, ε)

]ds+ o(εN) .

30


Let us take the vector norm || · || in Rm of this result and use the fact that f is Lipschitzin Ω w.r.t the first argument, c.f. (3.5),

||RN(t, ε)|| = ||∫ t

t0

[f(u(N)(s, ε) +RN(s, ε), s, ε)− f(u(N)(s, ε), s, ε)

]ds||+ o(εN)

≤∫ t

t0

||f(u(N)(s, ε) +RN(s, ε), s, ε)− f(u(N)(s, ε), s, ε)|| ds+ o(εN)

≤ Lf

∫ t

t0

||RN(s, ε)|| ds+ o(εN) , t0 ≤ t ≤ t0 + T .

(3.15)The size of ||RN ||(t, ε) can now be estimated from the following Lemma:

Lemma 2. (Gronwall) Suppose for t ∈ [t0, t0 + T ],

ϕ(t) ≤ a

∫ t

t0

ϕ(s) ds+ b (t− t0) + c ,

with ϕ(t) continuous, ϕ(t) ≥ 0 for t ∈ [t0, t0 + T ] and constants a > 0, b, c ≥ 0, then

ϕ(t) ≤( ba

+ c)ea(t−t0) − b

a

for t ∈ [t0, t0 + T ].

We now apply this Lemma to (3.15) with a = Lf = Os(1), b = o(εN) and c = o(εN)to obtain

RN = o(εN) for t0 ≤ t ≤ t0 + T ,

with T independent of ε. Hence we have proved that u(N) given in (3.14) is indeed anAE of the true solution uε of the IVP (3.4). To summarize, we have:

Theorem 1. Suppose that the problem P0 in (3.11) has a unique solution u0(t) in theinterval t0 ≤ t ≤ t0 + T . Moreover, suppose that the IVP denoted Pε in (3.4) is suchthat

i) f ∈ CN(Rm ×D × (0, ε0]),

ii) f(u, t, ε) is Lipschitz with respect to u in any bounded domain Ω ⊂ Rm,

iii) both f and the initial condition V have AEs as in (3.6) as ε→ 0.

Then we have the regular AE

uε(t) =N∑n=0

εnun(t) + o(εN) for t0 ≤ t ≤ T ,

where the un for n ≥ 1 satisfy the linear IVPs (3.12).

31


Remarks:

• Once the solution u0 of th reduced problem P0 is known, it is rather easy to increasethe accuracy of the AE by solving just a sequence of linear IVPs. This should ingeneral be much easier than solving the full problem Pε, and it shows the purposeof perturbation theory: breaking down a complicated problem into a sequence ofsimpler problems we are able to solve.

• The AE in Theorem 1 holds only for finite time intervals [t0, t0 + T ], where Tis independent of ε. If one wants to have approximations on longer time scalessuch as T ∼ 1/ε, one needs to resort to other techniques such as averaging or thetechnique of multiple scales.

3.3. Example: The nonlinear spring

For ε ∈ (0, ε0], consider the IVP (Duffing equation)

d2xεdt2

+ xε + εx3ε = 0 , t > 0 ,

xε(0) = 1 ,dxεdt

(0) = ε .

We can easily recast this problem into the form of (3.4) by defining uε = (uε,1, uε,2) viauε,1 := xε and uε,2 := dxε

dt, which leads to

Pε

d

dt

(uε,1uε,2

)=

(uε,2−uε,1

)+ ε

(0−u3

ε,1

),(

uε,1uε,2

)(0) =

(10

)+ ε

(01

).

(3.16)

Assuming uε = u0 + εu1 + . . . leads to the series of problems

P0

d

dt

(u0,1

u0,2

)=

(u0,2

−u0,1

),(

u0,1

u0,2

)(0) =

(10

),

P1

d

dt

(u1,1

u1,2

)=

(u1,2

−u1,1

)+

(0−u3

0,1

),(

u1,1

u1,2

)(0) =

(01

).

...

32


The solution of the reduced problem is x0(t) = u0,1(t) = cos(t). The computation of thesolution to the linear problem P1 is left as training to the students in the exercise classof this course. There, it will be shown that an AE of the exact solution to Pε , for finitetime 0 ≤ t ≤ T , can be given as

xε(t) = cos(t)− 3

8εt sin(t) + ε

[sin(t) +

1

8cos3(t)− 1

8cos(t)

]+ o(ε) . (3.17)

Remark that this is an AE only for T < ∞ independent of ε, due to the secular term−3

8εt sin(t). This term is not o(1) for t ∼ 1

ε, or more general for 1

ε t, and it blows up

for instance for t ∼ 1ε2

.

3.4. The method of the strained coordinate

In the preceding section we obtained the AE (3.17) for the solution of the Duffingequation, which is valid for finite times only as ε→ 0. The general question is whetherone can extend the validity of AEs stated in Theorem 1 to longer times, namely of theorder t ∼ 1

ε. Indeed, there are multiple techniques to do so, some of them we shall

discuss in this course on a fundamental basis: we start with the method of the strainedcoordinate, then introduce the method of averaging and finally briefly touch on thetechnique of multiple scales.

The method of the strained coordinate relies on a change of the independent variablet ∈ D to a new variable τ via

t = (1 + εω1 + ε2ω2 . . .) τ . (3.18)

Here, the ωi ∈ R are free parameters which will be chosen appropriately in order toremove the secular terms from the AE. This method has been developed by Lindstedtand Poincare for solving problems in celestial mechanics and is also called the Lindstedt-Poincare method [4]. Let us consider again the nonlinear spring as an example. Substi-tuting the transformation (3.18) into (3.16) yields

Pε

d

dτ

(u∗ε,1u∗ε,2

)= (1 + εω1 + ε2ω2 . . .)

[(u∗ε,2−u∗ε,1

)+ ε

(0

−(u∗ε,1)3

)],(

u∗ε,1u∗ε,2

)(0) =

(10

)+ ε(1 + εω1 + ε2ω2 . . .)

(01

),

where u∗ε(τ) = uε(t) via the change of variables (3.18). Let us assume again a regularansatz of the form

u∗ε(τ) = u∗0(τ) + εu∗1(τ) + . . . ,

33


then substituting and sorting in powers of ε leads to the series of problems

P0

d

dτ

(u∗0,1u∗0,2

)=

(u∗0,2−u∗0,1

),(

u∗0,1u∗0,2

)(0) =

(10

),

P1

d

dτ

(u∗1,1u∗1,2

)=

(u∗1,2−u∗1,1

)+

(ω1u

∗0,2

−(u∗0,1)3 − ω1u∗0,1

),(

u∗1,1u∗1,2

)(0) =

(01

).

...

From the reduced problem P0 we obtain u∗0,1(τ) = cos(τ) and u∗0,2(τ) = − sin(τ). Wesolve the problem P1 by using Duhamel’s formula and the homogeneous solution, writtenin terms of the “rotation matrix” (see the exercise class)

R(t, t0) :=

(cos(t− t0) sin(t− t0)

− sin(t− t0) cos(t− t0)

),

as (u∗1,1u∗1,2

)(τ) = R(τ, 0)

(01

)+

∫ τ

0

R(τ, s)

(ω1u

∗0,2(s)

−(u∗0,1(s))3 − ω1u∗0,1(s)

)ds . (3.19)

We are interested in particular in u∗1,1(τ), which corresponds to the first line in eq. (3.19):

u∗1,1(τ) = sin(τ)− ω1

∫ τ

0

cos(τ − s) sin(s) ds

−∫ τ

0

sin(τ − s) cos3(s) ds

−ω1

∫ τ

0

sin(τ − s) cos(s) ds .

(3.20)

The second integral in this equation yields the terms from the AE obtained previouslyin (3.17) (see the exercise class for details),

−∫ τ

0

sin(τ − s) cos3(s) ds = −3

8τ sin(τ) +

1

8cos3(τ)− 1

8cos(τ) . (3.21)

We still have the free parameter ω1 in order to cancel the secular term in (3.21). Forthis let us compute the remaining two integrals in (3.20). We realize that∫ τ

0

sin(τ − s) cos(s) ds =

∫ τ

0

sin(s) cos(τ − s) ds ,

34


and compute the remaining integral using

cos(τ − s) = cos(τ) cos(s) + sin(τ) sin(s) ,

which leads to∫ τ

0

cos(τ − s) sin(s) ds = cos(τ)

∫ τ

0

cos(s) sin(s) ds+ sin(τ)

∫ τ

0

sin2(s) ds

=1

2cos(τ) sin2(τ) + sin(τ)

1

2

[− sin(τ) cos(τ) + τ

]=

1

2τ sin(τ) .

Therefore, we obtain

−2ω1

∫ τ

0

cos(τ − s) sin(s) ds = −ω1τ sin(τ) .

In order to cancel the secular term in (3.21) we need to choose

ω1 = −3

8.

The result for u∗1,1 then becomes

u∗1,1(τ) = sin(τ) +1

8cos3(τ)− 1

8cos(τ) ,

which is free of secularities. In principle, this procedure could be carried out also foru∗2,1 and for even higher orders. Up to first order we obtained

x∗ε(τ) = u∗ε,1(τ) = cos(τ) + ε[

sin(τ) +1

8cos3(τ)− 1

8cos(τ)

]+ o(ε) ,

and

τ =t

(1− 38ε+ o(ε))

.

Here, the expansion of the denominator yields ωε := 1 + 38ε+ o(ε), which is the angular

frequency of the nonlinear spring, determined in the process. The solution of the originalproblem (3.16) is thus approximated as

xε(t) = x∗ε(ωεt) = cos[(

1 +3

8ε)t]

+O(ε) . (3.22)

The theory of averaging we discuss next will provide the rigorous justification that theAE (3.22) is indeed valid for long times of the order t ∼ 1

ε.

35


3.5. Averaging

In section 3.2 we dealt with problems of the form

duεdt

= f(uε, t, ε) , uε(t0) = V ∈ Rm , t ∈ D ⊂ R , (3.23)

and proved that a regular AE of the solution uε exist for finite times t0 ≤ t ≤ t0 + T ,with T < ∞ independent of ε, if f is sufficiently regular and can be expanded in apower series in ε (c.f. Theorem 1). In this section we deal with a method that isappropriate for constructing AEs of uε that are uniformly valid in a much longer timeinterval t0 ≤ t ≤ t0 + T/ε - this method is called averaging .

3.5.1. Generic example

Before we state the fundamental theorem of averaging, let us illustrate the idea by meansof a simple example. Consider a perturbed Hamiltonian system for the ”action-anglevariables” I and ϕ, given as the IVP

dI

dt= εg(ϕ) ,

dϕ

dt= ω , t > 0 ,

I(0) = I0 , ϕ(0) = ϕ0 .

(3.24)

Here, ω ∈ R is a given frequency and the function g(ϕ) is assumed continuous and2π-periodic in ϕ. The solution of this system reads

ϕ(t) = ϕ0 + ωt , I(t) = I0 + ε

∫ t

0

g(ϕ0 + ωs) ds . (3.25)

We estimate the integral to

ε

∫ t

0

g(ϕ0 + ωs) ds ≤ εtmaxϕ∈2π|g(ϕ)| ,

and thus, for finite times t ∼ 1, I(t) = I0 + O(ε), which is accordance with Theorem 1.However, we can do better. Let us introduce the average g and fluctuation g of g withrespect to the angle variable ϕ,

g :=1

2π

∫ 2π

0

g(ϕ) dϕ , g(ϕ) := g(ϕ)− g .

By construction the average of the fluctuation is zero, g = 0. We can now split theintegral in (3.25) into two parts and write the solution of (3.24) as

ϕ(t) = ϕ0 + ωt , I(t) = I0 + εt g + ε

∫ t

0

g(ϕ0 + ωs) ds

= I0 + εt g +ε

ω

∫ ωt

0

g(ϕ0 + s′) ds′

36


We see that the remaining integral is 2π periodic in ϕ,∫ ωt+2π

0

g(ϕ0 + s′) ds′ =

∫ ωt

0

g(ϕ0 + s′) ds′ +

∫ ωt+2π

ωt

g(ϕ0 + s′) ds′

=

∫ ωt

0

g(ϕ0 + s′) ds′ + g︸︷︷︸=0

=: h(ωt) .

Hence the function h(ωt) is bounded for t ∈ [0,∞) and

ϕ(t) = ϕ0 + ωt , I(t) = I0 + εt g +ε

ωh(ωt) .

It seems that the dynamics of I(t) can be split into two parts:

1. a ”systematic part” I0+εt g, which is O(1) for times 0 ≤ t ≤ Tε

with T an arbitraryconstant, and whose derivative is εg,

2. an ”oscillating part”, given by the periodic function εωh(ωt), which is O(ε) for

t ∈ [0,∞).

We deduce that for times 0 ≤ t ≤ Tε, the solution of the ”averaged problem”

dI

dt= εg ,

dϕ

dt= ω , t > 0 ,

I(0) = I0 , ϕ(0) = ϕ0 ,

is an AE to O(ε) of the solution of (3.24),

ϕ(t) = ϕ0 + ωt , I(t) = I0 + εt g +O(ε) , 0 ≤ t ≤ T

ε.

Here, the secular term has been isolated and given a clear physical meaning as thesystematic motion of the system, around which the true solution fluctuates.

3.5.2. K.B.M. Theorem

Let us now come from the example to the general case and state the main theoremof averaging. The averaging principle has first been proved by Krilov, Bogoliubov andMitropolski and is thus known as the KBM-method (or K.B.M. theorem). There areseveral variants of the theorem, depending on the properties of the vector field f in(3.23), for instance periodicity with respect to t.

Let us consider IVPs of the form

Pε

duεdt

= εf(uε, t) , t > 0 ,

uε(0) = V ∈ Rm ,

(3.26)

37


for ε ∈ (0, ε0]. Remark that the vector field is multiplied by ε, which is necessary forhaving existence and uniqueness of a solution in an interval of order t ∼ 1

εunder the

following assumptions on the vector field f :

i) f is T -periodic in t with T independent of ε,

ii) f is defined on Ω = Br(V )×[0,∞), with the ball Br(V ) = x ∈ Rm : ||x− V || ≤ r;it is continuous in Ω and continuously differentiable in Br(V ),

iii) f and its Jacobian ∂f∂x∈ Rm×m have in Ω norms which are bounded by a constant

M , independent of ε.

Under these assumptions we have

Theorem 2. Let uε denote the solution of the problem Pε as in (3.26), and further letu0 stand for the solution of

P ε

du0

dt= εf(u0) , t > 0 ,

u0(0) = V ∈ Rm ,

where

f(x) =1

T

∫ T

0

f(x, t) dt ,

and the integration is performed as if x is constant. Then

uε − u0 = O(ε) for 0 ≤ t ≤ t1ε,

where t1 ∼ 1.

Proof. Rescaling to the macroscopic time τ = εt, the Cauchy-Lipschitz (or Picard-Lindelof) theorem gives existence and uniqueness of uε in the interval 0 ≤ t ≤ d

Mε,

where M = maxΩ ||f(u, t)||, || · || being some vector norm in Rm. Moreover, sincemaxBr(V ) ||f(u)|| ≤ M , the unique solution u0 exists in the same interval and staysinside the ball Br(V ).

In what follows we shall denote by f(x, t) := f(x, t) − f(x) the fluctuation of thevector field f . Our strategy is to make a clever ansatz for the approximation of uε inthe form u(1) = u0 + O(ε) and to show that the difference uε − u(1) is of order ε on thetime scale t ∼ 1

ε. To achieve this we need to exploit the average and the fluctuation of

the vector field f . To get an idea, let us play around with the true solution uε in order

38


to derive an ansatz involving u0:

uε(t) = V + ε

∫ t

0

f(uε(s), s) ds

≈ V + ε

∫ t

0

f(uε(t), s) ds

= V + ε

∫ t

0

f(uε(t)) ds+ ε

∫ t

0

f(uε(t), s) ds

≈ V + ε

∫ t

0

f(u0(s)) ds+ ε

∫ t

0

f(u0(t), s) ds .

This is what we want, because we recognize the solution of the averaged problem,

u0(t) = V + ε

∫ t

0

f(u0(s)) ds .

Therefore, let us propose as an approximation to uε the function

u(1)(t) := u0(t) + ε h(t, u0(t)) ,

where we defined

h(t, u0(t)) :=

∫ t

0

f(u0(t), s) ds .

As usual, we now aim to derive an IVP for the error term uε − u(1) and prove that itstays small on a time scale of order t ∼ 1

ε. For this, note that

duεdt− du(1)

dt= εf(uε, t)− εf(u0)− ε∂h

∂t− ε∂h

∂x· du0

dt

= εf(uε, t)− εf(u0)− εf(u0, t)− ε2∂h

∂x· f(u0)

= ε[f(uε, t)− f(u(1), t)

]+ ε[f(u(1), t)− f(u0, t)

]− ε2∂h

∂x· f(u0) .

Integrating this equation, using the initial condition u(1)(0) = u0(0) = V , and takingthe vector norm yields

||uε(t)− u(1)(t)|| ≤ ε

∫ t

0

||f(uε, s)− f(u(1), s)|| ds+ ε

∫ t

0

||f(u(1), s)− f(u0, s)|| ds

− ε2

∫ t

0

∣∣∣∣∣∣∂h∂x· f(u0)

∣∣∣∣∣∣ ds≤ εLf

∫ t

0

||uε − u(1)|| ds+ εLf

∫ t

0

||u(1) − u0|| ds− ε2

∫ t

0

∣∣∣∣∣∣∂h∂x· f(u0)

∣∣∣∣∣∣ ds .

39


In the second line we used that f is Lipschitz in Br(V ) because it belongs to C1(Br(V ))and denoted the Lipschitz constant by Lf , independent of (t, ε). Moreover, we knowthat ||u(1) − u0|| = ε||h|| = O(ε) uniformly in [0,∞) because h is T -periodic. We alsoassumed that the norms of f and its Jacobian are bounded in Ω. This yields

||uε(t)− u(1)(t)|| ≤ εLf

∫ t

0

||uε − u(1)|| ds+ ε2t C , C = O(1) .

We can now apply Gronwall’s Lemma 2 with a = εLf , b = ε2C and c = 0 to obtain

||uε(t)− u(1)(t)|| ≤ εC

LfeεLf t − εC

Lf= O(ε) for 0 ≤ t ≤ t1

ε,

with t1 arbitrary of order one. Moreover, since h is bounded, also

uε − u0 = O(ε) for 0 ≤ t ≤ t1ε.

3.5.3. The standard form

The K.B.M-Theorem considers IVPs of the form

Pε

duεdt

= εf(uε, t) , t > 0 ,

uε(0) = V ∈ Rm ,

where the vector field f is multiplied by ε. This is called the standard form in the theoryof averaging. A technique called variation of the constants is often useful to transform aperturbation problem into the standard form. Consider the “full” perturbation problem

P fullε

duεdt

= f0(uε, t) + εf1(uε, t) , t > 0 ,

uε(0) = V ∈ Rm ,

The reduced problem

du0

dt= f0(u0, t) , u0(0) = V ∈ Rm , (3.27)

has as solution the flow map Φ0(t, V ), Φ0 : R+ × Rm → Rm, with Φ0(0, V ) = V (seeAppendix A for details on the flow). Variation of constants assumes that the solutionto the full problem P full

ε can be written as

uε(t) = Φ0(t, V (t)) .

Taking the time derivative yields

duεdt

=dΦ0(t, V )

dt=∂Φ0(t, V )

∂t+∂Φ0

∂x(t, V ) · dV

dt= f0(Φ0(t, V ), t) + εf1(Φ0(t, V ), t) ,

40


where ∂Φ0/∂x ∈ Rm×m is the Jacobian of the flow. Since Φ0 satisfies the reducedproblem (3.27), in case that ∂Φ0/∂x is invertible, we obtain an equation for V (t) in thestandard form,

dV

dt= ε

(∂Φ0

∂x(t, V )

)−1

f1(Φ0(t, V ), t) .

This equation can be very messy, however in some cases variations of the constants isvery useful. For quasilinear problems for instance, where f0(uε, t) = Auε in the reducedproblem, with A ∈ Rm×m a constant matrix, we obtain for the flow

Φ0(t, V ) = eAt V .

The standard form then becomes

dV

dt= ε e−At f1(eAt V, t) .

Clearly, if there is an eigenvalue of A with non-vanishing real part, then we might haveproblems with this equation even if f1 is bounded. Moreover, for f0(uε, t) = Auε+g(t) inthe reduced problem, with g an arbitrary continuous function, we have from Duhamel’sformula the flow

Φ0(t, V ) = eAt V +

∫ t

0

eA(t−s)g(s) ds . (3.28)

The corresponding standard form reads

dV

dt= ε e−At f1(Φ0(t, V ), t) ,

where Φ0 from (3.28) is to be inserted.

3.6. The method of multiple scales

For this we have no time. In case of interest check the literature list given for the course.

41

4. Singular perturbations of linearODEs

4.1. Generalities

When studying singular perturbations we are still concerned with initial/boundary valueproblems of the form (3.1). However, this time an ansatz of the form (3.3) will fail toproduce an AE of the solution that is uniformly valid in the domain D. Instead, we willbe faced with the emergence of so-called boundary layers in the vicinity of sub-manifoldsof D. These are regions where the solution uε shows strong variations, which eventuallylead to discontinuities as ε → 0. The prototypical example of such behavior can beobserved in the problem

εdx

dt ε+ xε = 0 , t > 0 , xε(0) = 1 .

The ε-family of solutions reads xε(t) = e−t/ε and we observe the formation of a boundarylayer at x = 0 as ε→ 0. A regular ansatz of the form xε = x0 +εx1 + . . . would moreoverfail to approximate the correct solution uniformly in [0,∞) as ε→ 0.

In the analysis of singular perturbations, the concept of a formal approximation playsan important role:

Definition 10. Suppose that Pε[uε] = 0 for ε ∈ (0, ε0], hence uε denotes the ε-family ofsolutions to problem (3.1). A formal approximation of order δ(ε) of uε in D ⊂ D isa function vε : D → Rm such that Pε[vε] = O(δ(ε)) uniformly in D.

This definition means that the formal approximation satisfies the equations and theinitial conditions in Pε up to an error of order O(δ(ε)). This does not mean that it isan AE of the solution uε. In order to prove that uε − vε = o(1) uniformly in D we stillneed to show that the error term is small. This is usually done by means of a-prioriestimates for the error, which satisfies some differential equation. The general strategyfor treating perturbation problems can be summarized as follows:

1. Find a formal approximation to the solution.

2. Come up with an a-priori estimate for the error and prove that it is small uniformlyin D.

In this course we will apply this strategy to a linear IVP and to a linear BVP whichfeature boundary layers and are thus singular perturbation problems. For such kind ofproblems, finding a formal approximation can be difficult and needs some more sophis-ticated methods than in the regular case.

42

CHAPTER 4. SINGULAR PERTURBATIONS OF LINEAR ODES

4.2. The initial value problem

In this section we will formulate a theorem regarding the AEs of solutions to the followinglinear IVP, for ε ∈ (0, ε0],

Pε

Lε[uε] = ε

d2uεdt2

+ a(t)duεdt

+ b(t)uε = f(t) , t > 0 ,

uε(0) = α ,duεdt

(0) = β ,

(4.1)

where α, β ∈ R are constants, independent of ε. Regarding the coefficient functions inLε we make the following assumptions:

i) a, b, f ∈ C1([0,∞)),

ii) a(t) ≥ a0 > 0 for t ≥ 0 with a0 independent of ε.

The second condition is particularly important in the subsequent construction of an AE.Setting ε = 0 in Pε yields the reduced problem, a first-order ODE which is ill-posedbecause of the two initial conditions which in general cannot be fulfilled simultaneously.Even though the perturbation is singular, we can expect to extract some useful informa-tion from the reduced problem. Neglecting for the moment the second initial condition,let us consider the problemL0[w0] = a(t)

dw0

dt+ b(t)w0 = f(t) , t > 0 ,

w0(0) = α .

(4.2)

The solution reads

w0(t) = α eq(t,0) +

∫ t

0

eq(t,s)f(s) ds , (4.3)

with

q(t, s) = −∫ t

s

b(s′)

a(s′)ds′ .

Due to our assumptions on a, the function q is well-defined. We also see the difficultywith the second initial condition,

dw0

dt(0) = − b(0)

a(0)α + f(0) 6= β in general .

We need another method to be able to incorporate the second initial condition. Inorder to resolve the boundary-layer region near t = 0, we will “zoom in” and define a“microscopic variable” ξ via

t = ενξ , ν > 0 .

In the new variable ξ the differential operator Lε reads

Lε = ε1−2ν d2

dξ2+a(ενξ)

ενd

dξ+ b(ενξ) .

43


Much like in the algebraic toy problems from the beginning of the course, let us apply theprinciple of dominant balance: choosing ν = 1/2 would lead to an unbalanced reducedproblem, such that ν = 1 is the only meaningful choice. We thus define the “microscopicoperator”

Lε :=L−1

ε+ L0 ,

with

L−1 =d2

dξ2+ a(0)

d

dξ, L0 = ξa′(ϑεξ)

d

dξ+ b(εξ) for some ϑ ∈ (0, 1) .

Here, we used the mean value theorem (Mittelwertsatz der Differentialrechnung) to write

a(t) = a(0) + ta′(ϑt) for some ϑ ∈ (0, 1) .

Our plan is that the solution of L−1[v] = 0 with appropriate boundary conditions de-scribes in some way the behavior of uε in the boundary layer near t = 0. Hence, weattempt a formal approximation of the form

uε(t) = w0(t) + v( tε

)+Rε(t) = w0(t) + v(ξ) +Rε(t) , (4.4)

where w0 is the solution of (4.2), Rε denotes the error term and v is the solution of theboundary value problem

L−1[v] =d2v

dξ2+ a(0)

dv

dξ= 0 , ξ > 0 ,

1

ε

dv

dξ(0) = β − dw0

dt(0) , lim

ξ→∞v(ξ) = 0 .

(4.5)

While the boundary condition at t = 0 ensures that our formal approximation (4.4)indeed satisfies the second initial condition of the original problem, the boundary con-dition at ξ → ∞ ensures that v is a boundary layer term with contributions only in asmall right neighborhood of t = 0. In particular, writing dv

dξ= r we obtain

r(ξ) = ε

(β − dw0

dt(0)

)e−a(0)ξ ,

which after integration in ξ yields

v(ξ) = ε

(dw0

dt(0)− βa(0)

)e−a(0)ξ + c ,

where c stands for the constant of integration. The second boundary condition in (4.5)then leads to c = 0 and thus

v( tε

)= ε

(dw0

dt(0)− βa(0)

)e−a(0) t

ε . (4.6)

We can now go on to prove

44


Lemma 3. Suppose that uε solves (4.1) under the assumptions stated, that w0 is thesolution of (4.2), and let v denote the solution of the microscopic problem (4.5). Thenw0(t) + v(t/ε) is a formal approximation of order O(ε) of uε(t), i.e.

Lε[w0 + v] = f(t) +O(ε) ,

and

(w0 + v)(0) = α +O(ε) ,d(w0 + v)

dt(0) = β +O(ε) ,

for t ∈ [0, T ] with T <∞ independent of ε.

Proof. Applying the differential operator Lε to w0(t) + v(t/ε) yields

Lε(w0 + v) = εd2w0

dt2+

1

ε

d2v

dξ2+ a(t)

dw0

dt+a(t)

ε

dv

dξ+ b(t)(w0 + v)

= εd2w0

dt2+

1

ε

d2v

dξ2+a(0)

ε

dv

dξ︸︷︷︸=0

−a(0)

ε

dv

dξ+ a(t)

dw0

dt+ b(t)w0︸︷︷︸

=f(t)

+a(t)

ε

dv

dξ+ b(t)v

= f(t) + εd2w0

dt2+a(t)− a(0)

ε

dv

dξ+ b(t)v .

It is easy to show that the last three terms in this expression are of order O(ε) on anybounded segment [0, T ] with T independent of ε. Since a and b and their first derivatives

are continuous, they are bounded in [0, T ]; then (4.3) implies that d2w0

dt2(t) is bounded

in [0, T ] and hence d2w0

dt2= O(1). Moreover, from (4.6) we obtain that v(t/ε) = O(ε)

uniformly in [0, T ] and hence bv = O(ε) uniformly in [0, T ]. For the remaining term weuse the mean value theorem to write

a(t)− a(0)

ε

dv

dξ

( tε

)=t

εa′(ϑt)

dv

dξ

( tε

)= εξa′(ϑt)

(β − dw0

dt(0)

)e−a(0)ξ .

We remark that a′ is uniformly bounded in [0, T ] and that ξe−a(0)ξ is uniformly boundedin [0,∞). As a conclusion we have

Lε(w0 + v) = f(t) +O(ε) .

With regards to the initial conditions we have, using (4.5) and (4.6),

(w0 + v)(0) = α + ε

(dw0

dt(0)− βa(0)

),

d(w0 + v)

dt=

dw0

dt+ β − dw0

dt= β .

which shows that w0 + v is indeed a formal approximation of uε in [0, T ], because itsatisfies the equations as well as the initial conditions up to order O(ε).

45


Using the above Lemma, we can set up an equation for the remainderRε = uε − w0 − vin (4.4):

Lε[Rε] = g(t, ε) , 0 < t ≤ T ,

Rε(0) = α1 ,dRε

dt= 0 ,

(4.7)

with

g(t, ε) = −εd2w0

dt2− a(t)− a(0)

ε

dv

dξ− b(t)v = O(ε) , α1 = ε

β − dw0

dt(0)

a(0)= O(ε) .

An a-priori estimate for the solution Rε of (4.7) and its derivative dRε

dtcan be be

obtained using the method of energy integrals. This method consists in deriving anestimate for the energy

Wε := m(R2ε + ε(R′ε)

2) ,

where m is some generic constant. Estimating the energy rather than the solution of anequation is useful if we are interested in bounds for the first derivative of the solution,in addition to bounds for the solution itself. In the present case we observe that the”microscopic term” v is of order O(ε), while its first derivative is of order O(1). Hencewe expect that the solution w0 of the modified reduced problem (4.2) is an asymptoticapproximation (AA) of uε in [0, T ] and that dw0

dt+ dv

dtis an AA of duε

dtin [0, T ]. We shall

now go on to show that this is indeed the case. For the energy we have

Lemma 4. Let a0 denote the lower bound of the function a(t) in the differential op-erator Lε described in (4.1). For ε ∈ (0, ε0] with a suitable constant ε0, the energyWε = a0

2(R2

ε + ε(R′ε)2) corresponding to the solution of problem (4.7) satisfies

Wε ≤M(||g||2[0,T ] + α21) e

2Ma0T,

where || · ||[0,T ] denotes the L2-norm and with

M = max[0,T ]

(| − 2b+ a′ + ε+ b2|, |a′ + 2 + a2|, 2, a(0)

),

independent of ε.

Proof. In what follows we shall write the equation in problem (4.7) as

Lε[Rε] = εR′′

ε + aR′ε + bRε = g . (4.8)

Let us assume the energy constant m to be m = a0/2. At first we estimate the energyto

Wε =a0

2(R2

ε + ε(R′ε)2)

≤ (a0 − ε)R2ε + ε(a0 − ε)(R′ε)2

≤ 2Wε + ε2RεR′ε

≤ a(R2ε + ε(R′ε)

2) + ε2RεR′ε

=

∫ t

0

d

ds

[a(R2

ε + ε(R′ε)2) + ε2RεR

′ε

]ds+Wε(0) .

(4.9)

46


The idea is to wright the integrand as h1Wε + h2 such that we can find ε-independentbounds for h1 and h2. The application of Gronwall’s Lemma will then lead us to thedesired estimate. Let us call the integrand Q and perform the differentiation:

Q :=[a(R2

ε + ε(R′ε)2) + ε2RεR

′ε

]′= a′R2

ε + 2aRεR′ε + εa′(R′ε)

2 + ε2aR′εR′′

ε

+ ε2(R′ε)2 + ε2RεR

′′ε .

Now we want to express the terms with the help of the equation (4.8). In order to getthe terms with the second derivatives we must multiply (4.8) with 2aR′ε and with 2Rε

and add up the results:

ε2aR′εR′′

ε + 2a2(R′ε)2 + 2abR′εRε = 2aR′εg

ε2RεR′′

ε + a2RεR′ε + 2bR2

ε = 2Rεg

+

⇒ ε2aR′εR′′

ε + ε2RεR′′

ε + a2RεR′ε

= −2a2(R′ε)2 − 2abR′εRε − 2bR2

ε + 2aR′εg + 2Rεg ,

⇒ ε2aR′εR′′

ε + ε2RεR′′

ε + a2RεR′ε +a′R2

ε + εa′(R′ε)2 + ε2(R′ε)

2︸︷︷︸terms 1,3,5 of Q

= −2a2(R′ε)2 − 2abR′εRε − 2bR2

ε + 2aR′εg + 2Rεg

+a′R2ε + εa′(R′ε)

2 + ε2(R′ε)2︸︷︷︸

terms 1,3,5 of Q

,

⇒ Q = (−2b+ a′)R2ε + (−2a2 + εa′ + 2ε)(R′ε)

2 − 2abR′εRε + 2aR′εg + 2Rεg(4.10)

In order to estimate the Q-term we use

2aR′εg ≤ a2(R′ε)2 + g2 ,

2Rεg ≤ R2ε + g2 ,

−2abR′εRε ≤ b2R2ε + 2a2(R′ε)

2 .

Inserting this into (4.10) yields

Q ≤ (−2b+ a′ + ε+ b2)R2ε + (εa′ + 2ε+ εa2)(R′ε)

2 + 2g2 .

Substituting this estimate for Q as well as Wε(0) = a(0)α21 into the integral in (4.9)

leads to

a0

2(R2

ε + ε(R′ε)2) ≤M

∫ t

0

(R2ε + ε(R′ε)

2) ds+M

∫ t

0

g2 ds+Mα21 , (4.11)

where

M := max[0,T ]

(| − 2b+ a′ + ε+ b2|, |a′ + 2 + a2|, 2, a(0)

)= O(1) in [0, T ] .

47


Applying the Gronwall Lemma 2 in (4.11) with a = 2M/a0, b = 0 and c = M(||g||2[0,t] + α21)

(this is the L2-norm of g) yields

a0

2(R2

ε + ε(R′ε)2) ≤M(||g||2[0,t] + α2

1) e2Ma0t ≤M(||g||2[0,T ] + α2

1) e2Ma0T.

With the aid of the above Lemma we immediately obtain bounds for |Rε| and for |R′ε|:

|Rε(t)| ≤√

2M

a0

(||g||[0,T ] + |α1|) eMa0T

t ∈ [0, T ] ,

|R′ε(t)| ≤1√ε

√2M

a0

(||g||[0,T ] + |α1|) eMa0T

t ∈ [0, T ] .

Using that g = O(ε) and α1 = O(ε) in [0, T ] we thus have

|Rε(t)|+√ε|R′ε(t)| = O(ε) in [0, T ] .

To summarise this section we have

Theorem 3. Let uε stand for the solution of

Pε

ε

d2uεdt2

+ a(t)duεdt

+ b(t)uε = f(t) , t > 0 ,

uε(0) = α ,duεdt

(0) = β ,

where α, β ∈ R are constants, independent of ε and

i) a, b, f ∈ C1([0,∞)),

ii) a(t) ≥ a0 > 0 for t ≥ 0 with a0 independent of ε.

Moreover, let w0 and v denote the solutions of the modified reduced problem (4.2) and ofthe microscopic problem (4.5), respectively. Under these conditions one has for ε → 0the asymptotic approximations

uε − w0 = O(ε) in [0, T ] ,

andduεdt− dw0

dt− dv

dt= O(

√ε) in [0, T ] ,

where T > 0 is arbitrary but independent of ε. Moreover,

duεdt− dw0

dt= O(

√ε) in [δ, T ] ,

where δ > 0 is independent of ε.

48


Remarks:

• If a(t) is strictly negative it can be shown that uε diverges as ε→ 0 and the wholeconstruction breaks down. Moreover, if a(t0) = 0 at some point t0 then the reducedproblem becomes singular at this point and complications arise. The point t0 iscalled a turning point .

• The boundary layer term v has no influence on the point-wise asymptotic approx-imation of uε. The term is however necessary to obtain a point-wise AA of thederivative of uε to order

√ε in [0, T ]. The function v is asymptotically smaller

than any power of ε away from t = 0.

4.3. The boundary value problem

Consult the literature list given for this course.

49

5. Macroscopic limits of kineticequations

5.1. Introduction

Some of the most complex phenomena in nature can be viewed as systems of inter-acting particles: the motion of gases, fluids, plasmas, cars in traffic or the behaviorof animal herds, bird swarms and school of fish, for example. It is thus of fundamentalinterest to develop an understanding of these systems via mathematical models. Becausethe number of particles is usually extremely large, the “microscopic” (in the sense ofatomistic) description in terms of the exact motion of every particle is not practicable,and moreover not necessary for studying large scale properties. Essential features of suchsystems can be revealed by inspecting statistical averages, which leads to “macroscopicmodels” in terms of densities in position space (x-space). The Euler equations and theNavier-Stokes equation are examples of macroscopic models. These equations containcoefficients like viscosity and heat diffusivity, which have to be derived from the atomisticpicture. In order to have a sound macroscopic theory, in accordance with microscopicmodels, we should be able to formulate the micro-macro transition as a perturbationproblem, where the solution of the micro problem converges to the solution of the macroproblem as ε → 0. The formulation and study of such limits is a quite formidable taskand has concerned scientists for many decades, see [6] for a short review.

A fruitful approach has been the introduction of an intermediate, or “mesoscopic”level, the so-called kinetic description. In the kinetic picture the particle system isdescribed in terms of densities in phase space (position-velocity or x-v-space). Thecentral quantity is the “distribution function” f(t, x, v), to be understood such thatf(t, x, v)dxdv is the density of particles at the point (x, v) ∈ R6 in phase space at time t.The governing equation for f is the famous Boltzmann equation:

∂f

∂t+ v · ∇xf + F (x, t) · ∇vf = Q(f) ,

f |t=0 = f0(x, v) .

(5.1)

Here, F (x, t) stands for an external force and Q(f) is the (nonlinear) collision operator.If long-range interactions between the particles are to be included, the external force isreplaced by a mean-field interaction which depends on f . We will not concern ourselveswith this additional complication in this course. The Boltzmann equation (BE) hasbeen extensively studied [2, 3] and has gained renewed attention with the advent ofsemiconductor physics in the computer industry, where the mesoscopic description is

50

CHAPTER 5. MACROSCOPIC LIMITS OF KINETIC EQUATIONS

the best compromise between complexity and feasibility. Its main feature is a collisionoperator which accounts for the short-range interaction of the particles. The collisionshave the effect that the entropy of the system is steadily decreasing - the statement of thefamous H-theorem - and the dynamics are irreversible. Today, most of the macroscopicmodels can be understood as limits of the Boltzmann equation in the highly-collisionalregime, so-called fluid limits. In what follows we shall give a brief overview of the(formal) techniques used in order to obtain such limits. A discussion about the derivationof the BE from the atomistic picture can be found in the above references.

5.2. The free transport equation

Before we start with the study of fluid limits of the BE, let us consider the free transportequation (FTE), obtained from the BE in the absence of collisions:

∂f

∂t+ v · ∇xf + F (x, t) · ∇vf = 0 ,

f |t=0 = f0(x, v) .

(5.2)

We call the solution (X, V ) of the ODEs

dX

ds= V , X(t) = x ,

dV

ds= F (X, s) , V (t) = v ,

the “characteristics” of the FTE. Supposing that F is sufficiently smooth, let us sayF ∈ C2 with bounded derivatives up to second order, the Cauchy-Lipschitz theoremguarantees a solution X(s; t, x, v), V (s; t, x, v). The application

Φst : (x, v) 7→ (X(s; t, x, v), V (s; t, x, v))

defines the flow of the characteristic equations. It is now easy to show that the FTE hasa unique solution of the form

f(t, x, v) = f0(X(0; t, x, v), V (0; t, x, v)) .

This follows from the fact that f is constant along the characteristics,

d

dsf(s,X(s), V (s)) = 0 . (5.3)

Indeed,

d

dsf(s,X(s), V (s)) =

∂f

∂s+

dX

ds· ∇xf +

dV

ds· ∇vf

=∂f

∂s+ V · ∇xf + F (X, s) · ∇vf .

51


In view of the FTE (5.2) we obtain (5.3). Integration of (5.3) between 0 and t thenyields

f(t,X(t; t, x, v), V (t; t, x, v)) = f(t, x, v) = f(0, X(0; t, x, v), V (0; t, x, v)) .

The flow Φst has the group property

Φt3t1 = Φt3

t2 Φt2t1 ,

and the inverse of Φst is given by Φt

s. Hence the dynamics of the FTE are reversible.Moreover, the flow Φs

t conserves phase space volume. This means that for any volumeΩ(t) that is advected by the flow,

Ω(t) = (x, v) ∈ R6 : (x, v) = Φt0(x0, v0) with (x0, v0) ∈ Ω0 ⊂ R6 ,

we haved

dt

∫Ω(t)

d3x d3v =d

dt

∫Ω0

∣∣∣∣det∂Φt

0

∂(x, v)

∣∣∣∣ d3x0 d3v0 = 0 .

This follows Liouville’s theorem

d

dtdet

∂Φt0

∂(x, v)= 0 ,

related to the fact the vector field (v, F (x, t)) of the characteristics is divergence-free.

5.3. Properties of the collision operator

The collision operator Q(f) in the BE (5.1) is a nonlinear integral operator and hard todeal with in actual applications. However, there are some essential properties of Q whichcan be distilled to write down simpler collision operators with the same features. Thisapproach is sufficient for understanding fluid limits. The are three essential propertiesof Q:

1. Conservation of mass, momentum and energy:∫Q(f)

1v|v|2

d3v = 0 ,

Here and in what follows, if no boundaries are indicated with an integral sign, theintegration is performed over the whole R3.

2. The equilibria are Maxwellian functions:

Q(f) = 0 ⇔ f =M :=n

(2πT )d/2exp

[−|v − u|

2

2T

], (5.4)

where n, T ∈ R+ (strictly positive) and u ∈ R3 are arbitrary and may depend on(t, x), and d stands for the dimension of the position space, usually 1 ≤ d ≤ 3.

52


3. H-theorem: the entropy S(t) is decreasing with time,

d

dtS(t) =

d

dt

∫ ∫f ln(f) d3x d3v ≤ 0 ,

andd

dtS(t) = 0 ⇔ f =M .

There are two well-known simplified forms of Q that feature all the essential properties:

• The BGK (Bhatnagar–Gross–Krook) operator:

QBGK(f) = ν(Mf − f) .

Here, ν > 0, independent of v, stands for the collision frequency and the MaxwellianMf is related to f via the velocity moments, hence in (5.4) we choose n, u, T as

n =

∫f d3v , u =

1

n

∫v f d3v , T =

1

nd

∫|v − u|2 f d3v . (5.5)

Since the Maxwellians M in (5.4) are Gaussians in v we have the following inte-grals: ∫

M

1v|v|2

d3v =

nnu

nT · d

.

For the Maxwellian Mf , from (5.5) it follows that

∫(Mf − f)

1v|v|2

d3v = 0 , (5.6)

such that conservation of mass, momentum and energy is guaranteed by the BGKoperator. The identification of the equilibria is trivial. In order to see that theH-theorem is verified, let us multiply the BE (5.1) by ln(f) and integrate over xand v to obtain

∂

∂t

∫ ∫f ln(f) d3x d3v = ν

∫ ∫(Mf − f) ln(f) d3x d3v .

Here the transport terms vanish due to the fact that we demand

lim|x|→∞

f(t, x, v) = 0 , lim|v|→∞

f(t, x, v) = 0 . (5.7)

Substituting on the left-hand-side the definition of the entropy S we can write

dS

dt= ν

∫ ∫(Mf−f) ln

(f

Mf

)d3x d3v+ν

∫ ∫(Mf−f) ln(Mf ) d3x d3v . (5.8)

53


We can write the logarithm of the Maxwellian as

ln(Mf ) = ln

(n

(2πT )d/2

)− |v − u|

2

2T= a+ b · v + c|v|2 .

Then, using (5.6) we obtain that the second term on the right-hand-side of (5.8)is zero. Moreover, note that that (x − y)(ln(y) − ln(x)) is negative for all valuesx 6= y and zero only for x = y, because the functions f(x) = x and f(x) = ln(x)are strictly increasing. It follows that

dS

dt≤ 0 ,

dS

dt= 0 ⇔ f =Mf .

• The Fokker-Planck-Landau operator:

QFPL(f) = ν∇v ·[(v − u)f + T ∇vf

].

Here, u and T are defined from f as in (5.5). It is straightforward to verifythe conservation of mass, momentum and energy (use integration by parts in v).Moreover,

∇vMf = −(v − u)

TMf

implies that QFPL(Mf ) = 0. On the other hand,

QFPL(f) = 0 ⇒ (v − u)f + T ∇vf = 0 ⇒ ∇v ln(f) = −(v − u)

T,

which implies that f =M. With regards to the H-theorem we have

dS

dt= ν

∫ ∫∇v ·

[(v − u)f + T ∇vf

]ln(f) d3xd3v

= ν

∫ ∫∇v ·

[(v − u)f + T ∇vf

]ln

(f

Mf

)d3xd3v

= −ν∫ ∫ [

(v − u)f + T ∇vf]Mf

f

(∇vf

Mf

− f

M2f

∇vMf

)d3x

= −ν∫ ∫

1

Tf

[(v − u)f + T ∇vf

]2

d3xd3v

≤ 0 .

Moreover, we see that ddtS = 0 if and only if (v− u)f + T ∇vf = 0, which leads to

the Maxwellian Mf .

54


5.4. Moment equations

The velocity moments of the distribution function f are defined as n(x, t)nu(x, t)w(x, t)

:=

∫ 1v|v2|/2

f(x, v, t) d3v .

Here, n is called the mass (density), nu stands for the momentum (density) and wdenotes the energy (density). These quantities will be called “fluid variables” inthe following. As an auxiliary quantity, the temperature T is defined via

d

2nT (x, t) :=

1

2

∫|v − u(x, t)|2f(x, v, t) d3v ,

where d is the dimension of the position space, here d = 3. It follows that theenergy can be expressed as

w =d

2nT +

1

2n|u|2 ,

which can be interpreted as the sum of an internal and a kinetic energy. Theequations of motion for the fluid variables follow directly by taking moments ofthe Boltzmann equation:

∂

∂t

nnuw

+∇x ·

nunu⊗ u+ P

wu+ P · u+Q

=

0nFnuF

.

Here, (u⊗ u)i,j = uiuj denotes the tensor product of two vectors, P stands for thepressure tensor and Q is the heat flux, defined by

P(x, t) :=

∫[v − u(x, t)]⊗ [v − u(x, t)] f(x, v, t) d3v ,

Q(x, t) :=1

2

∫|v − u(x, t)|2[v − u(x, t)] f(x, v, t) d3v .

The moment equations are thus not closed, i.e. they still depend on the fulldistribution function f via the pressure tensor and the heat flux. Asymptoticclosure via perturbation theory is about the computation of closure relations forP and Q, thereby expressing them in terms of the fluid variables n, nu and w.In order to obtain hydrodynamic equations, the perturbation problem consists offinding asymptotic expansions of fε as ε→ 0, solution of the BE

Pε

∂fε∂t

+ v · ∇xfε + F (x, t) · ∇vfε =Q(f)

ε,

fε|t=0 = fε,in(x, v) ,

55


with the boundary conditions (5.7). For the collision operator one usually chooseseither the BGK or the Fokker-Planck-Landau operator discussed above. Using aPoincare type ansatz (often called Hilbert ansatz in this context) of the form

fε = f0 + εf1 + . . .

and using the properties of the collision operator permits to compute the coefficientfunctions f0, f1, etc. Because collisions are of order O(1/ε), they dominate theother terms and the limit distribution limε→0 fε = f0 will be a Maxwellian. Theequations satisfied by the moments of f0 are the well-known Euler equations ofgas dynamics. Moreover, the equations for moments of f0 + εf1 turn out to be thecompressible Navier-Stokes equations (see Hausarbeit - Topic 2).

56

A. Fundamentals of initial-valueproblems

We present here some basic notions and theorems about systems of first-order ordinarydifferential equations with initial conditions. For an in-depth reading on IVPswe recom-mend the books [9, 11, 12].

A.1. Problem setting

In what follows J ⊂ R denotes an interval on the real line and and Ω ⊂ Rn stands foran open and connected subset of Rn. We shall be concerned with systems of ODEs ofthe form

dz(t)

dt= f(z(t), t) , (A.1)

where f : Ω × J → Rn is a suitable regular function. The image of f is called thedirection field of the system (A.1). A C1-curve z : J → Rn satisfying (A.1) is tangenteverywhere to the direction field. In case that f is independent of t, the system (A.1)is called autonomous . For autonomous systems, we call a function E : Ω → R a firstintegral if f(z) · ∇E(z) = 0 ∀ z ∈ Ω. It is easy to see that

d

dtE(z(t)) = 0 ⇐⇒ E is a first integral .

The level sets of E are hypersurfaces in Ω. In case that E is a first integral, E(z(t)) = c,the motion lies in the corresponding hypersurface.

If (A.1) is furnished with a condition z(t0) = V ∈ Ω for some t0 ∈ J we call it aninitial value problem (IVP):

dz(t)

dt= f(z(t), t) ,

z(t0) = V .

(A.2)

A function z is a solution of (A.2) if and only if it satisfies the integral equation

z(t) = V +

∫ t

t0

f(z(s), s) ds . (A.3)

Hence, (A.2) and (A.3) are equivalent formulations of the IVP. By a solution of (A.2)we mean a C1-curve z : I → Ω on some interval I ⊂ J , where t0 ∈ I, z(t0) = V and

57

APPENDIX A. FUNDAMENTALS OF INITIAL-VALUE PROBLEMS

(A.2) holds for all t ∈ I. We remark that a solution may be defined only on I ⊂ J , evenif the IVP is defined on J . The graph of a solution is the set

γz := (t, z(t)) ∈ J × Ω : t ∈ I . (A.4)

A proper extension of z : I → Ω, solution of (A.2), is a function z : I → Ω, whereI ⊂ I ⊂ J , I 6= I and z(t) = z(t) for t ∈ I. A maximal solution of the IVP is a solutionwith no proper extension. The corresponding interval I is called the maximal interval.

From the integral formulation (A.3) of the IVP we are inclined to search for solutionswith lesser regularity, namely z ∈ C0(I,Ω), such that the integrand on the right-hand-side of (A.3) is a piecewise continuous function of s. The integral solution than satisfies(A.2) at points of continuity of t 7→ f(z(t), t) due to the fundamental theorem of calcu-lus. Such solutions are important in case where the function f is not continuous withrespect to its second argument (sometimes arising in control problems).

A.2. Existence and uniqueness

Existence and uniqueness of solutions to the IVP (A.2) depend on the properties ofthe function f . Peano’s existence theorem states that there is at least one solution of(A.2) on some interval I ⊂ J if f is continuous on Ω × J . For uniqueness one needsan additional property: f must be locally Lipschitz with respect to its first argument,which means there exist neighborhoods Ωz ⊂ Ω of z and Jt ⊂ J of t such that

||f(z1, s)− f(z2, s)||||z1 − z2||

≤ L <∞ ∀ z1, z2 ∈ Ωz, s ∈ Jt .

The constant L, which possibly depends on Ωz and Jt, but not on z1, z2, is called theLipschitz constant. Remark that the limit ||z1 − z2|| → 0 in the above expression isfinite.

Example 13. Let us view the function f(z) = z1/3 as a mapping f : R → R. f iscontinuous but not locally Lipschitz. Consider the point z = 0. Since f(−1) = −1, onehas limε→0 |f(ε) − f(−ε)|/(2ε) = limε→0 ε

−2/3 = ∞, which means that f is not locallyLipschitz at z = 0. Moreover, let J = R = Ω and consider the scalar IVP

dz(t)

dt= t z1/3(t) , z(0) = 0 . (A.5)

We readily verify that z = 0 on R is a maximal solution, but so is the continuouslydifferentiable function

z(t) =

(t/√

3)3, t ≥ 0

0 , t < 0.

Hence, there are at least two maximal solutions to the IVP (A.5). The cause for this isthat f(z) = z1/3 is not locally Lipschitz at z = 0.

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In what follows we will call

Assumption A: f is continuous on Ω× J and locally Lipschitz with respect toits first argument, and t 7→ f(z(t), t) is jointly continuous for continuous z.

We can now state the basic existence and uniqueness theorem:

Theorem 4. Under the assumption A, the IVP (A.2) has a unique maximal solutionfor each (V, t0) ∈ Ω× J . The associated maximal interval is denoted I(V, t0) ⊂ J .

This theorem is known as the Cauchy-Lipschitz theorem or the Picard-Lindelof the-orem. There are two different approaches to the proof, depending on whether one usesthe continuity of f with respect to the first argument (Gronwall’s lemma, c.f. Theorem4.17 in [9] for example), or a fixed-point technique (c.f. Theorem 4.22 in [9] for example).

A.3. Transition map and local flow

Let us now introduce the notion of the transition map ψ of the IVP (A.2). The transitionmap is called the local flow in case that the IVP is autonomous. We define the mapψ : dom(ψ) ⊂ J × Ω × J → Ω by the property that t 7→ ψ(t;V, t0) is the solution ofthe IVP (A.2). Hence the transition map can be viewed as the solution of the IVP withdependence on the initial condition. The domain of ψ is

dom(ψ) = (t, V, t0) ∈ J × Ω× J : t ∈ I(V, t0) .

Definition 11. The system (A.2) is autonomous if J = R and if f does not depend ont.

For an interval I ⊂ R let us denote the interval shifted by t0 via

I + t0 := t+ t0 ∈ R : t ∈ I .

We then have the following Corollary to the above theorem:

Corollary 5. Let f : Ω× J → Rn satisfy assumption A.

1. Let (V, t0) ∈ Ω× J and let s ∈ I(V, t0). Then I(ψ(s;V, t0), s) = I(V, t0) and

ψ(t;ψ(s;V, t0), s) = ψ(t;V, t0) ∀ t ∈ I(V, t0) . (A.6)

2. Assume that the system is autonomous. Then, for arbitrary t0, s ∈ R and V ∈ Ω,I(V, t0) = I(V, s)− s+ t0 and

ψ(t+ s− t0;V, s) = ψ(t;V, t0) ∀ t ∈ I(V, t0) . (A.7)

59


Proof. 1. The curve z : t 7→ ψ(t;ψ(s; t0, V ), s) is the unique solution of the IVP

dz(t)

dt= f(z(t), t) , z(s) = ψ(s;V, t0) . (A.8)

However, the curve z : t 7→ ψ(t;V, t0) clearly satisfies (A.8) too, which according toTheorem 4 means z(t) = z(t) for all t ∈ I(ψ(s;V, t0), s) and the interval is maximal.Moreover, since z is the unique solution of

dz(t)

dt= f(z(t), t) , z(t0) = V ,

we obtain I(V, t0) = I(s,ψ(s, t0, V )).2. The curve z : t 7→ ψ(t;V, s) is the unique maximal solution of

dz(t)

dt= f(z(t)) , z(s) = V ,

defined on the interval I(V, s). The curve z(t) := z(t + s − t0) is thus defined onI(V, s)− s+ t0. It is easily verified that z satisfies

dz(t)

dt= f(z(t)) , z(t0) = V ,

and is hence defined on the maximal interval I(V, t0).

The property (A.6) is the semigroup property of the transition map; it leads to anintuitive interpretation of ψ as a dynamical propagator. The property (A.7) mirrors thetranslational invariance of autonomous systems, that is a translation (with respect totime) of a solution is also a solution. We have another Corollary:

Corollary 6. Let f : Ω × J → Rn satisfy assumption A. Then the relation on Ω × Jdefined by

(V, t0) ∼ (D, s) if s ∈ I(V, t0) and D = ψ(s;V, t0)

is an equivalence relation, and the graphs of solutions to initial conditions (V, t0) ∈ Ω×Jare the equivalence classes.

Proof. We need to prove the three properties defining an equivalence relation:

1. Reflexivity: Clearly t0 ∈ I(V, t0) and ψ(t0;V, t0) = V such that (V, t0) ∼ (V, t0).

2. Symmetry: Assume that (V, t0) ∼ (D, s), hence

I(D, s) = I(ψ(s;V, t0), s) = I(V, t0) =⇒ t0 ∈ I(D, s) ,

where we used Corollary 5 in the second equality. Moreover,

ψ(t0; D, s) = ψ(t0;ψ(s;V, t0), s) = ψ(t0;V, t0) = V .

It follows that (D, s) ∼ (V, t0), which proves symmetry.

60


3. Transitivity: Suppose that (V, t0) ∼ (D, s) and (D, s) ∼ (E, r), then from theabove it follows that

r ∈ I(D, s) = I(V, t0)

E = ψ(r; D, s) = ψ(r;ψ(s;V, t0), s) = ψ(r;V, t0)

=⇒ (V, t0) ∼ (E, r) ,

which proves transitivity.

By definition, if (V, t0) ∼ (D, s), the point (D, s) ∈ Ω×J lies on the graph of the solutionz with initial condition z(t0) = V , so that the graphs form the equivalence classes.

An equivalence relation partitions a set into pairwise disjoint subsets (the equivalenceclasses), whose union forms the whole set. Therefore, the above Corollary states that thegraphs of two solutions with initial conditions at (V, t0) and at (D, s) are either identicalor disjoint, i.e. different graphs do not intersect (cross).

A.4. Autonomous systems

Let us now focus on autonomous systems,

dz(t)

dt= f(z(t)) , z(0) = V , (A.9)

where f is locally Lipschitz on Ω. We may set the initial condition at t0 = 0 in full gen-erality because of the translation invariance of solutions, see the second part of Corollary5. A solution is a C1-curve z : I → Ω on some interval I containing 0, such that (A.9)holds for t ∈ I. Of course Theorem 4 applies in this case and gives existence and unique-ness of a maximally extended solution on a maximal interval IV = I(V, 0). Therefore,we may define the local flow ϕ of the system (A.9) as the map ϕ : (t, V ) 7→ ψ(t;V, 0).The domain of the flow is

dom(ϕ) = (t, V ) ∈ R× Ω : t ∈ IV .

In particular, t 7→ ϕ(t, V ) is the solution of the IVP (A.9). If IV = R for some V , thenthe solution t 7→ ϕ(t, V ) is said to be global. If IV = R for all V ∈ Ω, then ϕ is calledsimply the flow of (A.9), and ϕ is said to be a dynamical system. The local flow satisfies

ϕ(0, V ) = V ∀ V ∈ Ω ,

and for s ∈ IV : ϕ(t+ s, V ) = ϕ(t,ϕ(s, V )) ∀ t ∈ IV − s . (A.10a)

The first relation follows from the definition of ϕ. To prove the second one we evoke thesecond result of Corollary 5 with t0 = 0:

Iϕ(s,V ) = I(ψ(s;V, 0), 0) = I(ψ(s;V, 0), s)− s = I(V, 0)− s = IV − s .

61


Moreover, for t ∈ IV − s we get

ϕ(t,ϕ(s, V )) = ψ(t;ψ(s;V, 0), 0) = ψ(t+ s;ψ(s;V, 0), s) = ψ(t+ s;V, 0) = ϕ(t+ s, V ) .

Relation (A.10a) is termed the group property of the local flow. The terminology becomesclear when we regard the case where IV = R for all V ∈ Ω, so that ϕ is a dynamicalsystem. The family of mappings Φt : V 7→ ϕ(t, V ), t ∈ R, forms a commutative groupdue to Φs Φt = Φs+t for all s, t ∈ R.

For autonomous systems, the space Ω ⊂ Rn is called the phase space. In addition tothe graph of the solution z defined in (A.4), for autonomous systems one calls the imageof z an orbit (or trajectory) of the system. We have seen for general ODE systems thatdifferent graphs do not intersect. In the autonomous case, even more is true: differentorbits do not intersect. This is because the relation V ∼ V if V is in the orbit of V ,is an equivalence relation on Ω, similar to Corollary 6. The orbits are thus equivalenceclasses.

62

Index

asymptotic approximation, 15asymptotic convergence, 17asymptotic expansion, 4, 13, 16, 19asymptotic series, 16asymptotically equal, 19autonomous ODE, 22

boundary layer, 10

distinguished limit, 7dominant balance, principle of, 6dynamical system, 26

flow of an ODE, 24

gauge functions, 14graph of a solution, 23Gronwall lemma, 27

order function, 13

Poincare expansion, 20

reduced problem, 4, 11regular perturbation problem, 4

scaling, 8secular term, 4, 10, 12singular perturbation problem, 4

Taylor’s theorem, 17

63

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[6] Pierre Degond, Lorenzo Pareschi, and Giovanni Russo. Modeling and ComputationalMethods for Kinetic Equations . Modeling and Simulation in Science, Engineeringand Technology. Springer Science+Business Media, LLC, 2004.

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[8] Shi Jin. Efficient asymptotic-preserving (AP) schemes for some multiscale kineticequations. SIAM Journal on Scientific Computing, 21(2):441–454, 1999.

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