Shock Waves and Reaction—Diffusion Equations

Problems involving differential equations usually come in the following form: we are given an equation for the unknown function u, P(u) = f, on a domain Ω together with some “side” conditions on u. For example, we may require that u assumes certain preassigned values on ∂Ω, or that u is in L ²(Ω), or that u is in class C ^k in Ω. At first glance, it would seem that any of these extra conditions are quite reasonable, and that one is as good as the other. However, we shall see that this is far from being true, and that whichever additional supplementary conditions one assigns is intimately connected with the form of equation.

Roughly speaking, characteristics are curves which carry information. They are particularly relevant in the study of “initial-value” problems ; that is, in solving partial differential equations, in which the solution surface is required to assume prescribed values “initially.” Such a problem presupposes the existence of a distinguished coordinate, ξ, where the equation ξ, = 0 defines the “initial ” surface. Of course, as we have seen in the last chapter, one needs some kind of compatibility between the equation and the initial surface. The notion of characteristic serves to classify and make more precise these intuitive ideas.

In this chapter we study a simple but quite interesting equation for which the initial-value problem is well-posed. The ideas which we introduce here will be used in various places throughout the book, albeit at a “higher dialectical ” level. The equation is derived from physical considerations, and in the case we consider here, the solution u(x, t), may be thought of as describing the position of a vibrating string at a point x at a time t.

We shall extend the method of energy integrals to more general second-order (hyperbolic) operators. This “energy” method is a basic technique in the modern theory of partial differential operators, and in the course of our development, we shall establish some interesting and important classical inequalities.

There is a well-known theorem, called the Cauchy-Kowaleski theorem, which asserts that there exists a unique analytic solution of an analytic initial-value problem. Here, by an analytic initial-value problem, we mean a problem in which everything (the terms in the equation, the initial data, and the initial hypersurface), is analytic in a neighbourhood of a point (see [Ga]). The possibility is thereby left open as to whether there can exist a nonanalytic solution to this problem. Holmgren’s uniqueness theorem denies this possibility. We shall also find this result useful in Chapter 6 where we shall apply it to determine qualitative information on domains of dependence. For this reason, we shall prove a rather general version of the theorem.

We consider the equation for the homigeneous operator P: with initial data We assume that each a _a is constant, and that the hyperplane t = 0 is noncharacteristic with respect to P.

Consider the partial differential equation (in R ²), u _xy = f. If φ ∈ ₀ ² (R), then multiplying both sides of this equation by φ, and integrating by parts, gives ∫uφ _xy = ∫ fφ. Now suppose that u is not necessarily smooth, but that this last equation holds for each φ∈ C ₀ ^∞ (R ²). We could then say that f is the “weak” mixed derivative of u. We could actually go one step further and consider the linear functional on C ₀ ^∞ (R ²) defined by

Solutions of elliptic equations represent steady-state solutions; i.e., solutions which do not vary with time. They often describe the asymptotic states achieved by solutions of time-dependent problems, as t → ∞. Physically speaking, all the “rough spots” smooth out by the time this steady state is achieved.

In this chapter we begin to study nonlinear partial differential equations. The results which we obtain here all follow from the maximum principles which were obtained in Chapters 8 and 9. We shall show how they apply to nonlinear elliptic and parabolic partial differential equations. As a first application, we will use the strong maximum principles to prove comparison theorems ; i.e., pointwise inequalities between different solutions. These say, roughly, that if u and v are two solutions, and if u ≤ v on ∂D, it follows that u d v on D. Such theorems can be quite useful in obtaining qualitative information about solutions. For example, comparison theorems are often used to obtain information about the asymptotic behavior of solutions of parabolic equations as t→ + ∞. As a second application of the maximum principle, we shall show how it can be used to prove existence theorems. This is the method of “upper” and “lower” solutions, the solution being the limit of a monotone iteration scheme, where the monotonicity is a consequence of the maximum principle.

There is a well-known theorem in ordinary differential equations, going back to Poincaré, which states that the stability of a rest point can be inferred from “linearization.” More precisely, if one considers the ordinary differential equation in ℝⁿ, u′ = f(u), and ū is a rest point (so that/(ū) = 0), then if the differential (matrix) dfū) has all of its eigenvalues in the left-half plane, Re z x003C; 0, it follows that ū is asymptotically stable ; i.e., if u ₀ is near ū, then the solution of the equation through u ₀ tends to ū as t → + ∞. It is the main purpose of this chapter to prove an analogous theorem for partial differential equations, one which is sufficiently general to include systems of reaction-diffusion (parabolic) equations. In this context, the equation u′ = f(u) is replaced by an abstract equation of the form u _t = Au + f(u), where u takes values in a Banach space B; i.e., for each t, u(t) is in B, and A is a linear operator. The main example is the case where A is a linear elliptic operator. The “rest points” ū, in this setting now are solutions of the equation Au + f(u) = 0, and the linearized operator becomes A + df(ū). We shall show that if the spectrum of this operator lies in the left-half plane, then again one can conclude that ū is asymptotically stable. This leads us quite naturally to a study of the spectrum of such linear operators. We shall undertake this study in §A, and in §B we shall prove a linearized stability theorem. The techniques which we develop here will be applied to specific problems in later chapters. In §C we shall give a useful extension of a result obtained in §A ; this is the celebrated Krein-Rutman theorem

The invention of modern topology goes back to Poincaré, who was led to it in his study of the differential equations of celestial mechanics. Its development was taken over, for quite a while, by people who interestingly enough, seemed to have completely forgotten its origins. Perhaps this really was necessary in order that the subject develop rapidly. In any case, already in the twenties and thirties, people like Morse, Leray, Schauder, and others, were applying topological methods to differential equations, It is our purpose here to explain the relevance of some of these techniques to nonlinear differential equations.

Many problems in mathematics, and its applications to theoretical physics, chemistry, and biology, lead to a problem of the form where f is an operator on R x B ₁ into B₂, with B ₁ B ₂ Banach spaces. For example, (13.1) could represent a system of differential or integral equations, depending on a parameter λ. We are interested in the structure of the solution set ; namely, the set .

In recent years, systems of reaction-diffusion equations have received a great deal of attention, motivated by both their widespread occurrence in models of chemical and biological phenomena, and by the richness of the structure of their solution sets. In the simplest models, the equations take the form where u ∈ ℝⁿ, D is an n x n matrix, and f(u) is a smooth function. The combination of diffusion terms together with the nonlinear interaction terms, produces mathematical features that are not predictable from the vantage point of either mechanism alone. Thus, the term DΔu acts in such a way as to “dampen” u, while the nonlinear function f(u) tends to produce large solutions, steep gradients, etc. This leads to the possibility of threshold phenomena, and indeed this is one of the interesting features of this class of equations.

In this chapter we shall begin the study of quasi-linear systems of the form where, and. We assume that the vector-valued function f is C ² in some open subset Ω c ℝⁿ. These equations are commonly called conservation laws in analogy to the examples of such systems which arise in physics ; see the examples below.

In this chapter we shall obtain precise mathematical results on the existence and uniqueness of solutions for a single conservation law. In addition we shall also study the asymptotic behavior of our constructed solution. The existence problem will be attacked via a finite-difference method. Thus we shall replace the given differential equation by a finite-difference approximation depending on mesh parameters Ax and At. For every such pair (Δx, Δt) we shall construct a solution of the finite-difference equation, and we shall then obtain estimates which enable us to pass to the limit as the mesh parameters tend to zero in a certain definite way. The estimates which we obtain will be in the sup-norm and in the total variation-norm of the approximants, both sets of estimates being independent of the mesh parameters. It is worth noting that we are forced into obtaining bounds on the variation of the approximants, rather than (the usually encountered) bounds on derivatives, since the latter bounds would imply via the standard compactness criteria, that the limit would be continuous ; we know that this is not generally true.

In earlier sections, we have written the equations of gas dynamics in several different forms (see equations (18.3) and (18.3)′ of this chapter, and Examples 1 and 2 in Chapter 15). In many standard texts, it is shown that they are all equivalent for classical solutions ; i.e., they determine the same classical solutions. What we shall do here first is to prove that the computation of eigenvalues, Riemann invariants, and the genuine nonlinearity or linear degeneracy of the characteristic fields, are all independent of our choice of equations. That is, they are invariant under coordinate changes.

We consider a general system of conservation laws where u = (u ₁,⋯,u _n), with initial data The system (19.1) is assumed to be hyperbolic and genuinely nonlinear in each characteristic field, in some open set U ⊂ ℝⁿ (see Definition 17.7). We let λ₁(u) < ⋯ < λ_n(u) denote the eigenvalues of df(u). Concerning u _o(x), we assume that T.V.(u ₀) is sufficiently small, where by T.V.(-) we mean the total variation. With these assumptions, we shall show that the above problem has a solution which exists for all t > 0.

There are better results known for pairs of conservation laws than for systems with more than two equations. We have already seen an example of this in the last chapter ; namely, the interaction estimates are stronger when n = 2 than when n> 2. This was due to the existence of a distinguished coordinate system called Riemann invariants, which in general exists only for two equations. We shall study the implications one can draw using these coordinates. It turns out that the equations take a particularly nice form when written in terms of the Riemann invariants, and using this we can prove that for genuinely nonlinear systems, global classical solutions generally do not exist. (We only know this now for a single conservation law ; see Chapter 15, §B.)

We have studied second-order quasi-linear parabolic systems in Chapter 14, where it was assumed that the equations admitted a bounded invariant region. For the gas dynamics equations with all of the dissipative mechanisms taken into account (viscosity and thermal conductivity), and for various models of these, there may exist invariant regions, but they are usually unbounded. Thus we cannot conclude that the solution is a-priori bounded, and global existence theorems become more difficult to prove. One way to overcome this problem is to obtain “energy” inequalities in the unknown function and its derivatives, in a manner somewhat analogous to what we have done for linear hyperbolic equations in Chapter 4. In order to obtain these estimates for nonlinear equations, certain additional restrictions must be imposed : small data, special forms of the equations, restrictions on the data at infinity, and so on.

In any theoretical investigation of a real physical system, one is always forced to make simplifying assumptions concerning the true nature of the system. Since such idealizations are inevitable, it is reasonable to inquire as to how far one can go in this direction and still obtain satisfactory results. In certain cases, for example the motion of the planets, the equations of celestial mechanics provide a quite accurate model of the real physical system. In other situations, such as ecological or chemical interactions, or the study of large scale atmospheric phenomena, one either writes down certain reasonable relations between the quantities involved and their rates of change, or one tremendously reduces the number of actual equations involved. If such “leaps of faith” are to be of any use, it is necessary to study “rough” equations in “rough” terms. This in a nutshell, is our aim in this chapter and the next one. In other words, we want to fit these vague notions into a precise mathematical framework.

In this chapter we shall consider the Conley index from a more general point of view, one which allows us to apply the theory to a wide variety of equations including in particular, systems of reaction-diffusion equations. For such equations, it is not at all clear that the equations even define a flow. To get around such problems, we introduce the concept of a local flow and develop the theory in this setting. Roughly speaking, a local flow is a subset of the underlying space which is locally invariant for positive time; one thinks of a subspace of a function space, say L ₂, which is invariant under the equations for small t > 0.

The Conley index is a double-edged sword : if it is ever shown to be nontrivial, then this implies the existence of an orbit which stays in the isolating neighborhood for all time ; in this sense it gives an existence theorem. On the other hand, being a Morse-type index, it also carries stability information concerning the isolated invariant set. In this chapter we shall illustrate both of these properties for a special class of solutions of partial differential equations called travelling waves.