Lecture Introduction

Transcription

1 Lecture Introduction The theory of Partial Differential Equations (PDEs) is central to mathematics, both pure and applied. The main difference between the theory of PDEs and the theory of Ordinary Differential Equations (ODEs) is that an ODE involves a function (and its derivatives) of only one independent variable, while in PDE we deal with a function and its derivatives in several independent variables. This difference implies that the geometry related to the PDEs is much more complicated that in the case of ODEs. First examples of partial differential equations appeared already at the time of Isaac Newton ( ) and Gottfried Wilhelm Leibniz ( ), but a more systematic study was initiated by Leonhard Euler ( ). In the seventeenth and eighteenth centuries mathematicians and physicists realized that PDEs model many real-world problems. They were interested in boundary value problems involving vibrations of strings (in violins, harps) and columns of air (in organ pipes), arising in mathematical theories of music. The earliest contributors to such theories include Brook Taylor ( ) and Daniel Bernoulli ( ). Another example, Newton s mechanics, approximates some physical problems and this approximation is sufficient in our real-life here on Earth. However, going to a bigger scale, like solar system, galaxies, universe, Newton s mechanics is not good enough. In that case Einstein s relativity theory approximates the reality much more precisely. Naturally the mathematical description of real phenomena using differential equations, like all mathematical models, is an idealization. Even if we have an exact solution of an PDE, it only approximates a real-world problem. However in most applications such approximation is sufficient and in some cases (like quantum electrodynamics) astonishingly precise. Until the twentieth century, the theory of PDEs was considered as a branch of physics. However, their most striking attribute is their universality in applications ranging from analysis, differential geometry, functional analysis, probability theory, fluid or solid mechanics, electromagnetism and other branches of physics, biology, probability up to finanse. Moreover the trend of applying PDEs everywhere is increasing, partially due to the use of computers, which create models by discretization and approximation. Partial Differential Equations now form a big research area as a part of mathematics, but of course it is hard to separate it from applications especially from physical ones. A partial differential equation is a relation between independent variables, say x 1,..., x n, a function u = u(x 1,..., x n ) and its partial derivatives u xi1...x ik up to certain order, say k, F (x 1,..., x n, u, u x1,..., u xn,..., u xi1...x ik ) = 0. (1) The order of the equation is the highest order of derivative appearing in a non-trivial way (i.e. such that it cannot be removed by simple algebraic operations) in the equation. The general definition of partial differential equation has no practical meaning. This is due to the fact that in general, it is a difficult problem to find solutions of a partial differential equation. We wish to emphasize that the general theory of PDEs does not exist. Practically any (non-trivial) PDE requires its own theory and the theory can be very complicated. During this one-semester course we will study some particular classes of PDEs. We will practically skip the theory of PDEs of the first order (some examples will appear during the exercise course). This theory is important but it does not transfer to PDEs of higher orders and the main reason to omit it during this course is the lack of time. Instead, we will concentrate mostly on PDEs of second order and we will learn basics of the modern theory which was developed during the first half of the twentieth century. 1

2 The classical solution to a given partial differential equation of order k is a function u which belongs to the space C k and satisfies the equation in every point of its domain. The modern approach to the problem of solving PDEs is based on the idea of looking for solutions to PDEs in functional spaces different than classical spaces C k. This requires appropriate reformulation of a given PDE. Usually for each equation or a (not very wide) class of equations an appropriate space is specified in which solutions are considered. Moreover, these spaces often consist not only of functions, and some notions of generalized functions (e.g. distributions) are needed. In time, we will introduce the notion of the Sobolev spaces and the so-called weak solution to a PDE. We will start, however, with the classical solutions to some particular equations (Laplace s and Poisson s equations) and study their properties. 1.2 Notation By a domain in R n we mean an open and connected subset of the vector space. Assume that u is a function defined on a domain Ω R n. We say that u is of class C k (Ω), k = 0, 1, 2,..., if the function u and all partial derivatives up to order k they are continuous. We use a standard notation for derivatives: if u = u(x 1,..., x n ), then the derivative with respect to x j will be denoted by u x j or u xj or D xj u, j = 1,..., n Respectively, second order derivatives If α N n, α = (α 1,..., α n ) then we denote 2 u x j x k or u xj x k or D xj x k u. α = α i = α α n and D α u = i=1 α α. 1 x 1 x αn n By Du or u we denote a gradient of a function u with a norm If k = 2, 3,... then D k u = {D α u} α =k, ex. and The Laplace operator is defined as u(x) = Du(x) = (u x1 (x),..., u xn (x)) Du = ( n u xi 2) 1/2. i=1 u x1 D 2 x 1... u x1 x n u = u xnx1... u xnxn D k u = ( α: α =k D α u 2) 1/2. u = i=1 2 u x 2 i = T rd 2 u. 2

3 If Ω R n and F : Ω R n is a vector function F = (F 1,..., F n ), then divf = i=1 F i x i, and we see that u = div( u). 1.3 Examples Example 1.1 (Eikonal equation). Consider the following equation in two variables ( ) u 2 + x ( ) u 2 = 1 y in a region Ω R 2 outside a closed convex curve Γ = Ω, together with a boundary condition u = 0 on Γ. Solutions of the equation have a very nice geometric meaning, namely (up to a constant) they represent the distance function to the curve Γ. Example 1.2 (Cauchy Riemann system of equations). We know from the theory of analytic functions that a complex valued function f : C C, f = u+iv is complex differentiable (holomorphic) if and only if its real and imaginary parts satisfy the following system of differential equations u x = v y u y = v x. So one can say, in a sense, that the whole theory of holomorphic functions is an analysis of solutions to one particular system of differential equations of the first order (with constant coefficients). Example 1.3 (Equation of minimal surfaces). A minimal surface z = u(x, y) is a surface having least area for a given contour. It satisfies the second order quasi-linear equation ( ) ( ) u x u y + = u u 2 x Example 1.4 (Heat equation). The heat equation, also known as the diffusion equation, is an equation of the form u t x u = 0. It describes in typical applications the evolution in time of the density u of some quantity as heat, chemical concentration, etc. It appears as well in the study of Brownian motion. Example 1.5 (Laplace equation). The Laplace equation is an equation of the form u = 2 u x u 1 x = 0. n Solutions to this equation are called harmonic functions. The Laplace equation comes up in a wide variety of physical context. In a typical interpetation u denotes the density of some quantity (e.g. chemical concentration, temperature, electrostatic potential in a region without electric charge, gravitational potential) in equilibrium (stationary state). It also arises in the y 3

4 study of analytic functions. The real and imaginary parts of any complex analytic function f = f(z) on C, i.e., a function that locally can be expanded into a convergent power series in the complex variable z = x + iy, are harmonic functions in R 2. For instance, are harmonic functions in R 2. Re(z 2 ) = Re((x + iy) 2 ) = x 2 y 2 Im(z 2 ) = Im((x + iy) 2 ) = 2xy Example 1.6 (Wave equation). The wave equation is an equation of the form u tt c 2 x u = 0. The equation is a simplified model for a vibrating string (n = 1) or a membrane (n = 2). In the most interesting case (from the physical point of view), for n = 3, it can describe electromagnetic or acoustic waves or vibrations of an elastic solid. In these physical interpretations u(x, t) represents the displacement in some direction of the point x at time t Classification of equations The class of expressions we can write down and call partial differential equations is very wide (we could see several examples of it). As we have said before there is no general theory of PDEs, there are, however, some classes of PDEs featuring similar properties. We say that a partial differential equation is: linear if the function F in the equation (1) is linear with respect to u and all its derivatives, i.e. the PDE can be written in a form a α (x)d α u(x) = f(x), (2) α, α k where a α and f are given functions. We say that the equation is homogeneous equation if f 0 and nonhomogeneous otherwise. semi-linear if F is linear with respect to derivatives of the highest order and coefficients of the highest derivatives depend only on x, i.e. the equation can be written in a form a α (x)d α u(x) + a 0 (x, u(x), Du(x),..., D k 1 u(x)) = 0, (3) α, α =k where a α and a 0 are given functions. quasi-linear if F is linear with respect to derivatives of the highest order and the equation can be written in a form α, α =k a α (x, u(x), Du(x),..., D k 1 u(x)) D α u(x) + where a α and a 0 are given functions. a 0 (x, u(x), Du(x),..., D k 1 u(x)) = 0, (4) nonlinear or fully nonlinear if F is not linear with respect to the derivatives of the highest order. 4

5 1.4.1 Second order (semi-)linear equations Now consider a semi-linear second order partial differential equation i.e. an equation which are linear with respect to second order derivatives. Such equations can be written in a form 2 u a ij (x) + f(x, u, u x1,..., u xn ) = 0, (5) x i x j where u is unknown function (of a class C 2 (Ω)), a ij (x) = a ij (x 1,..., x n ), i, j = 1,..., n are given real valued continuous functions defined on a domain Ω R n, and f is a given real valued continuous function defined on Ω for the variable x and a suitable domain with respect to u and its derivatives. We can assume without loss of generality that a ij = a ji. Indeed, if we put then a ij = 1 2 (a ij + a ji ), a ij = 1 2 (a ij a ji ); a ij = a ji, a ij = a ji, a ij = a ij + a ij, and, because of the symmetry of second order partial derivatives of u, we have a ij u xi x j = a iju xi x j + a iju xi x j } {{ } =0 = a iju xi x j. Now, fix a point x Ω. Let λ 1,..., λ n be the eigenvalues of the matrix a(x) = (a ij (x)),...,n. Since we assume that a(x) is symmetric, all the eigenvalues are real: λ i R. Denote n + (x) = #{λ i > 0} n (x) = #{λ i < 0} n 0 (x) = #{λ i = 0}. Definition 1.1. We say that the semilinear second order partial differential equation (5) is elliptic at a point x, if n + (x) = n or n (x) = n; hyperbolic at x, if either n + (x) = n 1 and n (x) = 1 or n + (x) = 1 and n (x) = n 1; ultra-hyperbolic at x, if n + (x) + n (x) = n and 1 < n + (x) < n 1; parabolic in the broad sense at x, if 1 n 0 (x) n 1; parabolic in the narrow sense (or simply parabolic) at x, if n 0 (x) = 1 and either n + (x) = n 1 or n (x) = n 1. Equation (5) is said to be elliptic, hyperbolic, etc. throughout the entire region Ω if it is respectively elliptic, hyperbolic, etc. at every point of the region. We note that for the matrix a(x) we always have n 0 n 1. The above cases fill out all the possibilities. In our examples above, the heat equation is parabolic, the Laplace equation elliptic, and the wave equation hyperbolic (the three equations were the only semilinear ones among the examples). 5

6 The important fact is that the type of an equation doesn t change with a diffeomorphic change of variables. More precisely, assume Ψ : Ω U is a diffeomorphism of a class C 2 between two regions Ω and U in R n. Let u C 2 (Ω). Denote v = u Ψ 1 C 2 (U). Differentiating both sides of the equation u(x) = v(ψ(x)) we obtain and so u xi (x) = u xi x j (x) = µ=1 µ,ν=1 v yµ (Ψ(x)) y µ x i v yµ y ν (Ψ(x)) y µ x i y ν x j + µ=1 v yµ (Ψ(x)) 2 y µ x i x j, a ij u xi x j (x) = ã µν (y)v yµ y ν (y) µ,ν=1 (y = Ψ(x)) + lower order terms with derivatives v yµ, where ã µν (y) = y ν x j (x)a ij y µ x i (x) (x = Ψ 1 (y)). In other words, if à = (ã µν) and A = (a ij ), and J is the Jacobi matrix of the diffeomorphism Ψ, J = ( y µ x j ), then à = J A J T (calculated in appropriate points). The matrix of coefficients of second order derivatives transforms in the same way as a matrix of a quadratic form and therefore n 0 (Ã) = n0 (A) n + (Ã) = n+ (A) n (Ã) = n (A) which means that the type of the equation stays unchanged. 1.5 Initial and boundary conditions A partial differential equation, in general, has infinitely many solutions. The same, as you can recall, happens for ordinary differential equations. Many PDEs come from practical problems and then it is clear that a solution should satisfy some additional conditions. These conditions are motivated by physics and they came in two varieties: initial conditions and boundary conditions. What kind of conditions to impose mainly depends on the type of equation, like for the second order equations which we classified. Initial conditions. Assume that one of the independent variables corresponds to time (denote it by t) and other independent variables are spacial coordinates. An initial condition specifies the physical state at a particular time t 0. For example for the heat equation in 2-space variables u t (x, y, t) u xx (x, y, t) u yy (x, y, t) = 0 (x, y) R 2, t t 0 the condition u(x, y, t 0 ) = g(x, y), 6

7 where g is a given function, gives the initial temperature g(x, y) at the time t 0. Boundary condition. A boundary condition is a condition on the solution restricted to the boundary of the domain where the equation is defined. For example, for a vibrating string fixed at the endpoints the equation for the displacement is with the boundary condition u tt (x, t) u xx (x, t) = 0 a x b, u(a, t) = u(b, t) = 0 for all t. The three most important kinds of boundary conditions are: Dirichlet s condition: u is fixed on the boundary of the region on which it is defined; Neumann s condition: the normal derivative u n is fixed on the boundary of the region; Robin s condition: u n +au is specified on the boundary. Here a is a functions that depends on the boundary points of the domain and possibly on time t. By a boundary value problem we mean a given PDE with specified boundary conditions. The following three basic types of boundary value problems for linear second order differential equations may be distinguished: Cauchy s problem for equations of hyperbolic and parabolic type: initial conditions are given, the region coincides with the whole space R n and boundary conditions are absent; Boundary value problem for equations of elliptic type: boundary conditions on the boundary Ω of Ω R n are given, the initial conditions are absent; Mixed problem: for equations of hyperbolic and parabolic type: boundary conditions are given, Ω R n ; initial conditions and 1.6 Well posed problems Roughly speaking, a partial differential equation problem is said to be well-posed if it has a solution, that solution is unique and it only changes by a small amount in response to small changes in the input data. The first two criteria are reasonable requirements of a sensible model of physical situation, and the third one is often expected on the basis of experimental observations. This concept was first introduced by Jacques Salomon Hadamard ( ). A well-posed problem consists of a PDE in a domain together with a set of initial and/or boundary conditions satisfying the following natural requirements: Existence: the solution must exist within a certain class of functions from which the solution is chosen; Uniqueness: the solution must be unique within a certain class of functions from which the solution is chosen; Stability: the solution must depend continuously on the data of the problem (initial and boundary data, inhomogeneous term, coefficients of the equation, etc.). If the data changes a little, then the solution changes a little. This can be described precisely in mathematical terms. Some remarks about well-posed problems: 7

8 Too many or too few conditions. The choice of initial and boundary conditions depends on the equation. If the number of conditions is insufficient, then usually the solution is not unique and does not describe the related physical problem. If too many conditions are imposed, then it may happen that the problem has no solutions. Choice of class of functions. Also it should be noted that in the well-posedness it is important to specify not only the class of functions from which the data is taken, but also the space in which we are looking for solutions. The choice of class of functions determines the tools we can use. Also, choosing a too small space might result in nonexistence of a solution, too large in nonuniqueness. Stability - importance in approximation. When thinking of well-posedness, we must also remember that it is often impossible to find explicit solutions to problems of practical interest, so that approximation schemes, and in particular numerical solutions, are of vital importance in practice. Hence, the question of well-posedness is intimately connected with the central question of scientific computation in partial differential equations: given the data for a problem with a certain accuracy, to what accuracy does the computed output of a numerical solution solve the problem? Ill posed problem also can happen. Although many well-founded mathematical models of practical situations lead to well-posed problems, phenomena that are seemingly unpredictable, or at the least extremely sensitive to small perturbations, are not uncommon. Examples include turbulent fluid flows described by the Navier-Stokes equations. Pure and applied mathematicians must therefore be prepared for both well-posed and ill-posed partial differential equation models. Example 1.7 (Hadamard s Example of an ill-posed problem). We consider the Laplace equation in the half plane of R 2 : Ω = {(x, y) R 2 : x > 0} with the conditions u = 0 u(0, y) = 0 u (0, y) = n e n sin ny. x It can be shown (we will see this later) that the solution u of this problem is unique, for instance in the class C 2 for x 0. It is easy to check that the function u n (x, y) = e n e nx sin ny satisfies the problem. However we see that e n n sin ny is as small as we like if n is sufficiently large, but the solution u n (x, y) can be very large at some (x, y) if n is large (in the supremum norm). The problem does not meet the requirement of stability, since zero data gives trivial solution u 0. 8