INTRODUCTION

The heat or diffusion equation

models the heat flow in solids and fluids. It also describes the diffusion ofchemical particles. It is also one of the fundamental equations that haveinfluenced the development of the subject of partial differential equations(PDE) since the middle of the last century. Heat and fluidflow problems are important topics in fluid dynamics. Here theheat flow is combined with a fluid flow problem and the resulting equationis termed as energy equation. We begin with a derivation ofone-dimensional heat equation, arising from the analysis of heat flow in athin rod. Further, equation (8.1) is also a prototype in the class ofparabolic equations and hence the importance of studying this equation.

Derivation of One-Dimensional Heat Equation

Consider a thin rod of length L and place it along thex-axis on the interval [0; L]. Weassume that the rod is insulated so that its lateral surface is impenetrableto heat transfer. We also assume that the temperature is the same at allpoints of any cross-sectional area of the rod. Let denote, respectively, themass density, heat capacity and the coefficient of (internal) thermalconductivity, of the rod. Let us analyze the heat balance in an arbitrarysegment [x1; x2] of the rod, with𝛿x = x2 −x1 very small, over a time interval[t, t + 𝛿t];𝛿t small (see Figure 8.1).

Let u(x, t) denote thetemperature in the cross-section with abscissax, at time t. According toFourier's law of heat conduction, the rate ofheat propagation q is proportional to , with

S denoting the area of the cross-section. Here,

where 𝛿Q is the quantity of heat that has passedthrough a cross-section S during a time𝛿t. Thus,

Applying Fourier's law at x1 andx2, we obtain

Thus, the quantity of heat that has passed through the small segment[x1, x2] of therod, during time 𝛿t is given by

This influx of heat during time 𝛿t was spent inraising or lowering the temperature of the rod by𝛿u, say.

The first-order equations with real coefficients are particularly simple tohandle. The method of characteristics reduces the givenfirst-order partial differential equation (PDE) to a system of first-orderordinary differential equations (ODE) along some special curves called thecharacteristics of the given PDE. This will, in turn,help us to prove the existence of a solution to the Cauchyproblem or initial value problem (IVP)associated with the PDE. Complications do arise in case of quasilinear ornon-linear equations resulting only in local existence; thegeometry of the characteristics also becomes more involved and nonuniquenessof (smooth) solutions may also result. To motivate the ideas we begin by asimple example.

Example 3.1. Consider the transport equation in two independentvariables t and x, namely

for t 0, x ∈ ℝ, wherec > 0 is a given constant. This is a linear,first-order PDE. Consider the curve x =x(t) in the (x, t)plane given by the slope condition. These are straight lines with slope1/c and are represented by the equationx − ct =x0, where x0 is the pointat which the curve meets the line t = 0 (see Figure3.1(a)). These curves, straight lines in this case, are called thecharacteristic curves or simply thecharacteristics of (3.1). When c is afunction of t and x, the characteristiccurves need not be straight lines.

Now restrict the solution u(x, t) to acharacteristic x(t) = ct+ x0, that is, consider the function ofone variable U(t) =u(x(t,t). By the chain rule, it is easy to see that

using (3.1). Therefore, U ≡ constant and thusu is constant along the characteristicx − ct =x0. This observation can be used to solvethe IVP for the PDE (3.1) as follows:

Suppose the initial values of u are given on the linet = 0, that is, u(x,0) = u0(x) is given andu0 is a C1function. Now, for any point (x, t), t> 0 in the upper half plane, draw the characteristicpassing through the point (x, t). It is easy to see thatthis is given by the line with slope 1/c meeting theline t = 0 at the point (x –ct, 0). As shown above, we haveu(x, t) =u(x – ct, 0) =u0(x –ct).

Throughout the earlier chapters, we have seen the necessity of definingsolutions in a sense other than the well-known classical solutions. In otherwords, we need to go out of the realm of smooth solutions to capture thephysically relevant solutions. In fact, we have established the existence ofcertain types of weak solutions in the context of Hamilton–Jacobiequations and Conservation Laws. Further, we have also indicated in Chapter7, the existence of an integral formulation corresponding to a minimizationproblem given by the Dirichlet functional. Generically, we term suchsolutions as weak solutions. In fact, there are differentconcepts of weak solutions that have been developed in the last 100 years orso; like distribution solutions, transposition solutions, entropysolutions, viscosity solutions, and so on. Among them thenotion of weak solutions in the sense of distributions, weak formulation andSobolev spaces took centre stage in the first half of the last century, andsubsequently changed the scenario of the study of partial differentialequations. As remarked earlier, the necessity of introducing the aboveconcepts was evident from the study of Dirichlet functionals from the secondhalf of the eighteenth century and early part of the nineteenth century.Sobolev was quite successful in defining appropriate function spaces (theso-called Sobolev spaces) using generalized functions. A stable foundationwas given later by the introduction of distributions by L.Schwartz in the 1940s (see Brezis, 2011; Kesavan, 1989; Schwartz, 1966).This new development was not only useful in the study of PDEs, but it couldalso rigorously establish the notions of Dirac 𝛿function, its derivatives and the symbolic calculus developed by thephysicists including Paul Dirac.

Without going much into the details, we would like to introduce here thenotion of weak derivative of functions that are, otherwise,not classically differentiable and certain associated spaces, which arerequired to study solutions of weak formulation. In the process, we see someideas about the modern theory of PDEs.

WEAK DERIVATIVES

Quite often, given a function f, the valuef(x) represents a physical quantity atthe point x, say, temperature at x. Pointbeing a mathematical concept, physically (or experimentally), it cannotdetermine any physical quantity exactly at a point, rather we can onlyobtain the average quantity in a neighborhood of the point.

INTRODUCTION

Many physical laws, for example, conservation of mass, momentum and energy,occur as conservation laws. Dynamics of compressiblefluids, both in one and three dimensions, is a rich source of conservationlaws and offers many problems that are challenging; see, for example,Courant and Friedrichs (1976), Morawetz (1981), Whitham (1974), and Majda(1985). This subject has been one of the most active research fields, bothin the theoretical and computational aspects, for the past more than sixdecades. Because of its important applications in the aerospace engineering,the interest in this field is only growing. Yet, certain theoretical aspectsin the study of systems of conservation laws (e.g., Euler's equationsof gas dynamics) have not been resolved satisfactorily and remain achallenge. The interested reader will find some advanced topics in Smoller(1994), Majda (1985), and Lax (1973), and the references therein.

A conservation law asserts that the rate of change of the totalamount of a substance (e.g., a fluid) in a domain Ω inspace (an interval in the one-dimensional case) is equal to itsflux across the boundary𝜕Ω of the domain Ω. Ifu = u(x, t) denotesthe density of the substance at time t andf the flux, the conservation law is expressed as

where v denotes the outward unit normal on𝜕Ω and dS is thesurface measure on 𝜕Ω. The integral on theright-hand side is the total amount of the outflow of the substance across𝜕Ω, hence the negative sign. Assumingthat u and f are smooth, we obtain usingthe divergence theorem that

SinceΩis an arbitrary domain, by shrinking it to a point, we obtainthe following first-order PDE

satisfied by the density u and the flux f.Equation (5.2) is the conservation law expressed in thedifferentiated form and equation (5.1) in theintegral form.

Note that (5.1) is equivalent to

for all t1 and t2 witht1< t2.

In the present chapter, we consider only a single conservation law in one(space) dimension.

INTRODUCTION

In this chapter, we consider equations with analytic coefficients and discussthe existence and uniqueness of their solutions. Historically, theCauchy–Kovalevsky Theorem (CKT) is one of the first results in thetheory of partial differential equations (PDE) that addressed the questionof existence and uniqueness of solutions. Its proof introduced the conceptsof estimates that are at the heart of the modern PDE techniques. In fact,these estimates are known as a priori estimates for thesolution and its derivatives, derived before establishing the existence of asolution. More precisely, assuming the existence of a solution, suchestimates are derived. Thus, a priori estimates arenecessary conditions for the existence of a solution. The strategy is to usea priori estimates to define a suitable class offunctions in which a solution is sought. The rapid development of modernfunctional analysis provided impetus to the study of PDE and in the currentscenario, the study of PDE may be termed as advanced or applied functionalanalysis.

We first discuss the CKT for linear equations and then its generalization toa system of linear equations. Many of the books on the subject deal withthis classical theorem. We cite here John (1978), Hörmander (1976),Trèves (2006), and Folland (1995), among others. We follow theprocedure in Hörmander (1976) very closely for linear equations andTrèves (2006), Caflisch (1990), for linear systems.

We begin with a discussion of analytic functionsu(z) of n complex variablesz =(z1,…,zn) ∈ℂn. We use the followingnotations throughout this chapter: Let z0∈ ℂn and let u bea complex-valued function defined in a neighborhood ofz0. The function u is saidto be analytic at z0 ifu has the power series representation

where the power series converges absolutely in a neighborhood ofz0. Here, denotes a multi-index withnon-negative integers and. It follows

immediately from (11.1) that u is infinitely differentiablein a neighborhood of z0 and

Here Da denotes the differential operator of order and. Inparticular, for each j = 1,2,…, n, the function of onecomplex variable

where zj’s are held fixed, is analytic ina neighborhood of z0j.

INTRODUCTION

The wave equation

models many real-world problems: small transversal vibrations of a string,the longitudinal vibrations of a rod, electrical oscillations in a wire, thetorsional oscillations of shafts, oscillations in gases, and so on. It isone of the fundamental equations, the others being the equation of heatconduction and Laplace (Poisson) equation, which have influenced thedevelopment of the subject of partial differential equations (PDE) since themiddle of the last century.

We shall now derive equation (9.1) in the case of transverse vibrations of astring. Physically, a string is a flexible and elastic thread. The tensionsthat arise in a string are directed along a tangent to its profile. Weassume that the string is placed on the x-axis, with itsend points at x = 0 and x =L (not shown in the figure); see Figure 9.1.

We consider small transversal vibrations of the string, so that the motion ofthe points of the string is described by a functionu.(x; t), which givesthe amount that a point of the string with abscissa x hasmoved at time t. We also assume that the length of elementMM′ of the string corresponding tox and x +Δx; Δx being very small,is equal to Δx. We also assume that the tension ofthe string is uniform and denote it by T.

Consider a small element of the string corresponding to the abscissa pointsx andx+Δx. ForcesT act at the end points of this element along thetangents to the string. Let the tangents make angles𝜙 and 𝜙 +Δ𝜙 with x-axis, atM and M′, respectively. Then,the projection on the u-axis of the forces acting on thiselement will be equal to

Since we are assuming 𝜙 is small, we use theapproximation sin 𝜙 = tan𝜙 and obtain

Next, let be the linear density, that is, mass per unit length, of thestring. Applying the Newton's second law of motion, to the smallelement of the string under consideration, we Obtain

Dividing by Δx throughout and putting, results in(9.1).

In Chapter 3, we have briefly introduced the Hamilton–Jacobi equation(HJE)

as an example of a first-order equation to derive the characteristic curves,which form the well-known system of Hamilton's ordinary differentialequations (ODE). We also have seen there an example of a minimizationproblem, where the minimum value (known as the valuefunction) need not be differentiable at some points. However,at the points where it is differentiable, the value function does satisfyHJE. Hence, it is essential to look for functions outsidethe class of smooth functions, while attempting to solve the HJE andinterpret the solution in a generalized sense. Uniqueness is also an addedissue here.

The modern theory of distributions and Sobolev spaces provides a way tointerpret the derivatives in a weak sense. The distributiontheory is essentially a linear theory though it can also beapplied to certain non-linear problems where the equation is in theconservation form like, in conservation laws, or in a variational form.There is also an attempt to extend the calculus of distributions to handlenon-linear problems using the paradifferential calculus developed by Bony;see Hörmander (1988, 1997) and the references therein. In general,fully non-linear partial differential equations (PDE) quite often cannot behandled via distribution theory. See the sample problems in examples (3.19),(3.20). Crandall, Lions and Evans (see Crandall and Lions, 1981, 1983;Crandall et al., 1984, Lions, 1982) have initiated the theory ofviscosity solutions in 1980s to study non-linear PDE ina general framework for dealing with non-smooth value functions arising inthe dynamic optimization problems, like optimal control and differentialgames. Even a brief discussion of viscosity solutions is not included herebecause of technicalities involved.

In this chapter, we consider a very particular case of HJE, where an explicitformula for the solution will be derived. This is known asHopf–Lax formula for the value function. Ofcourse, the value function need not be smooth, but it satisfies certainspecial properties. Using these special properties, we define a weakernotion of a solution. We also present a uniqueness result under this newnotion of a solution, to capture the physically relevant non-smoothsolution.

We ventured into writing this book Partial DifferentialEquations knowing very well that writing a textbook on a veryold discipline, that too for beginners, is indeed a formidable task. Thisexercise was partly due to the good response we received for our first book,Ordinary Differential Equations, co-authored with RajuK. George, whose contents were also classical. The venture was also partlydue to the suggestions we have received during our interactions withstudents and teachers from various institutions in the country. The choiceof the contents for this book are largely based on such interactions andalso on our training in the subject. It is our wish that such a course onpartial differential equations (PDE) should seriously be taught at seniorundergraduate or beginning graduate level at various institutions in thecountry, so as to prepare a student for a more serious study of the advancedtopics.

This book should be accessible to anyone with sound knowledge in severalvariable calculus, save for a couple of chapters where the reader isexpected to have knowledge of the modern integration theory. The bookessentially deals with first-order equations, the classical Laplace andPoisson equations, heat or diffusion equation and the wave equation. Thefull generality was never on our minds. Numerical analysis and computationsare not considered here. Nevertheless, students and researchers working onthese aspects of the subject can also gain something from the book. Almostall the topics considered here, of course, arise from the realworldapplications in physics, engineering, biology, and so on. Though there is nodiscussion on the applications in the book, the community of students andresearchers from these applied fields can also benefit from the book. Wehave also presented a detailed description of the classification of PDE,including a motivation behind classification.

A few words about the title. The subject of PDE has undergone great changeduring the last 70 years or so after the development of modern functionalanalysis, in particular distribution theory andSobolev spaces. In the modern concept, the PDE isvisualized in a more general setup of functional analysis, where we look forsolutions in a sense weaker than the usual classical senseto address the more physically relevant solutions.

MULTIVARIABLE CALCULUS

Introduction

We plan to briefly introduce the calculus onℝn, namely the concept of totalderivative of multivalued function,. We are indeed familiar with the notionof partial derivatives. In the sequel, we will introduce the importantconcept of total derivative and discuss its connection tothe partial derivatives. We remark that the total derivative (known also asFrechét derivative) can be extended to infinitedimensional normed linear spaces, which is used in the analysis of morecomplicated problems especially arising from optimal control problems,calculus of variations, partial differential equations, and so on.

Motivation: One of the fundamental problems in mathematics (andhence in applications as well) is the following: Let f :ℝn →ℝn. Given y∈ ℝn, solve the system ofequations

f(x) = y (2.1)

and represent the solution as x =g(y) and if possible find goodproperties of g, namely its smoothness. More generally, iff : ℝn+m→ ℝn; x ∈ℝn; y ∈ℝm, solve the implicit system ofequations

f(x; y) = 0 (2.2)

and represent the solution as x =g(y). Consider the one-dimensionalcase, where f : ℝ → ℝ which isC1. Suppose that for somea. Then, by the continuity of, we see that in aneighborhood interval I of a.Hence preserves the sign in I, fis monotonic in I andf(I) is an interval. Thus, iff(a) = b, then theabove argument shows that f(x) =y is solvable for all y inf(I), a neighborhood ofb. This is the local solvability that is obtained bythe non-vanishing property of the derivative of f ata. This immediately shows the importance ofunderstanding the derivatives in the solvability of algebraic equations. Weremark that the mere existence of all partial derivatives does not guaranteethe local solvability. We need the stronger concept of total derivative.

Linear Systems: Let us look at the well-known linear system

Ax = y, (2.3)

where A = [aij] is a given n × n matrix.That is f(x) = Ax. Thesystem (2.3) can be rewritten as

The system (2.3) or (2.4) is uniquely solvable for x interms of y if and only if det A ≠ 0(global solvability).

INTRODUCTION

In this chapter, we study the wave equation in higher (space) dimensions andanalyze different problems associated with it: the Cauchy problem (initialvalue problem [IVP]), initial-boundary value problem in half-space, and soon. As noted in the introductory chapter, the wave equation arises in manyphysical contexts and it is a fundamental equation that has influenced theanalysis of solutions of general hyperbolic equations and systems. Unlikethe heat equation, the nature of solution of the wave equation depends onbeing the (space) dimension odd or even, except for one-dimensional case.This is also the reason to take up separately the study of the wave equationin higher dimensions. We also learn that the solution of the wave equationin even dimensions may be obtained from the solution in odd dimensions, bythe method of descent. We begin with the following Cauchyproblem for the homogeneous wave equation in the free spaceℝn:

Here n ≥ 2 is an integer, the (spatial)dimension, c > 0 is a constant, thespeed of propagation and u0;u1 are given smooth functions, the initial values. We describetwo methods to find a formula for the solution of (10.1) and (10.2).

The general references for this chapter are Ladyzhenskaya (1985), Rauch(1992), Mitrea (2013), Pinchover and Rubinstein (2005), McOwen (2005),Trèves (2006), Courant and Hilbert (1989), John (1971, 1975, 1978),DiBenedetto (2010), Renardy and Rogers (2004), Prasad and Ravindran (1996),Salsa (2008), Mikhailov (1978), Benzoni-Gavage and Serre (2007), Evans(1998), Kreiss and Lorenz (2004), and Vladimirov (1979, 1984).

To get an idea how this method works, we first consider a special case.Suppose the functions u0 andu1 are radial functions,that is,

THREE-DIMENSIONAL WAVE EQUATION: METHOD OF SPHERICAL MEANS

To get an idea how this method works, we first consider a special case.Suppose the functions u0 and u1 are radial functions,that is,

where. Also, extend u0 andu1 for by definingu0(−r =u0(r) andu1(−r) =u1(r). In this situation,we can expect a solution u of (10.1) also to be radial inx, that is u(x, t) =u(r, t). For such a functionu, we have

Therefore, (10.1) reduces to

Consider the case n = 3. Then, the functionv = ru satisfies the one-dimensionalequation

with initial conditions and, as follows from (10.3).

GENERAL NATURE OF PDE

It is no exaggeration to state that partial differential equations (PDE) haveplayed a vital role in the development of science and technology, primarilysince the beginning of the twentieth century. In the earlier stage, PDE weremainly used to describe physical phenomena, like vibrations of strings, heatconduction in solids, transport phenomena, to mention a few. Later, with theadvantage of mathematical modelling, the scope of using PDE for thedescription of phenomena occurring in biology, economics and even sociologybecame prominent.

Since the days of Newton or even earlier, many have attempted to describephysical processes using mathematics. Such a mathematical description oftenleads to linear differential, integral and even integro-differentialequations. Thus, a large number of PDE naturally come from mathematicalphysics. The initial developments in PDE, though, were mainly geared towardsobtaining solutions to a particular physical or engineering problem, it wassoon realized that many of the problems will have common features andsimilarities. This naturally led to the grouping of PDE that can be tackledin a single framework. This automatically leads to the abstraction of thesubject and the theoretical analysis that follows, hence, becomes moreimportant. This is one of the features we try to follow in the present book.Indeed, unlike ordinary differential equations (ODE), all PDE including thelinear ones cannot be treated in a single theoretical framework, leading tothe necessity of a classification. In fact, due to the diverse nature ofphysical phenomena, we remark that we cannot classify all the PDE.Nevertheless, a fairly good classification is available for the second-orderequations and interestingly a large number of physical and other problemslead to second-order equations. Also, for the three important classes ofequations, namely elliptic, hyperbolic and parabolic, general theories havebeen developed.

As mentioned above, a wide class of physical problems is described bysecond-order linear differential equations of the form Here the variablex varies in an open set in the physical spaceℝn; n = 1;2; 3 and the coefficients aij;bi and c are known from thephysical process; u is the unknown function andf denotes an external quantity, if any, influencing thephysical process.

INTRODUCTION

In this chapter, we discuss the classification of partial differentialequations (PDE), concentrating mostly on linear equations or equations withlinear principal part. In the modern theory of the subject, especially sincethe middle of the last century, the question of classification has takenaltogether new directions, which we only mention very briefly in the sectionon Notes. Also, the classification of PDEs is in general not complete. Thesecond-order equations in two variables have a fairly completeclassification, which is our main topic of discussion in this chapter. Webegin by a discussion on a Cauchy problem for a second-order linear PDE andhighlighting certain subtle differences with a Cauchy problem for anordinary differential equation (ODE). This naturally motivates towards thestudy of the classification of PDE. The general references for this chapterare Courant and Hilbert (1989), Rubinstein and Rubinstein (1998), Koshlyakovet al. (1964), John (1978), Mikhailov (1978), Ladyzhenskaya (1985), McOwen(2005), Renardy and Rogers (2004), Prasad and Ravindran (1996), Vladimirov(1979, 1984), and Hörmander (1976, 1984), among many others.

CAUCHY PROBLEM

We begin by describing a Cauchy problem or aninitial value problem (IVP) for a linear PDE andcomparing it with an IVP associated to a linear ODE. For simplicity of theexposition, we consider the second-order linear equation in a regionΩ in ℝn:

where the coefficients aij; ai; a andthe function f are given real (smooth) valued functionsdefined in Ω. The Cauchy problem for a second-order linear ODE

where the coefficients b; c and the functiong are smooth functions defined in an interval inℝ, consists in finding a solution u satisfying theinitial conditions and for somex0 and arbitraryu0, u1.

A Cauchy problem (or an IVP) associated with the PDE (6.1), in which initialconditions for u and its normal derivative are assigned onan (n − 1)-dimensional surface Γ in Ω.We will see that, in general, this may lead to a lack of existence oruniqueness depending on the nature of Γ.

INTRODUCTION

The reasons for studying Laplace and Poisson equations are twofold.Primarily, these equations arise in a wide variety of physical contexts.Secondly, as mentioned earlier, Laplace operator is a prototype of a verygeneral class of linear elliptic operators. In fact, the Laplace operatorpossesses many features of the general class of elliptic operators. Recallthat the most general form of second-order linear partial differentialequations (PDE) in n variables is given by

where x ∈ Ω; an open set inℝn,aij = aji. Theoperator L is said to be uniformlyelliptic if there exists an a > 0 suchthat for all. Recall that is the characteristic form associated with theoperator L. The ellipticity condition here implies that thecharacteristic variety is an empty set at all points in Ω.

Thus, the condition requires the uniform positive definiteness of thesymmetric matrix [aij(x)]. Ifwe take bi = ci =d = 0 and

there results the Laplace operator Δ. The ultimate interest is tostudy the classical solutions of the equationLu=f for a given dataf. The study of Laplace equationΔu=0 and Poisson equationΔu = f (potentialtheory) gives a starting point for the general theory ofLu = f. The Schaudertheory provides a general theory for Lu =f when the coefficients are smooth, that is,Hölder continuous and it is essentially an extension of the potentialtheory. The crucial result is the derivation of estimates (known asa-priori estimates), say, of the form: Anyu ∈ C2(Ω), asolution of Δu = f in adomainΩ ⊂ℝn satisfies a uniform estimate:

where Ω′ ⊂⊂Ω; 0 < a < 1 andC is a constant depending only on a.Here is the standard Ḧolder space.

Such estimates eventually (with delicate analysis) lead to the solvability ofthe equation. The above estimate is an interior estimateand we also require boundary/global estimates that depend on the smoothnessof the boundary as well.