Even if the connection between well posedness and prediction and confirmation is not entirely straightforward well posed problems are important because they are mathematically quite tractable. It is impossible to give here even a brief overview on the mathematical literature;
the list of references for the 1983 non-historical survey for linear partial differential equations, Fattorini (1983), is over a hundred pages long, while ill posed inverse problems themselves have their own journal and book series. Thus we are only going to point out few mathe-matical advantages of assumptions of continuous dependence; the reason why existence and uniqueness of a solution is mathematically desirable is rather clear.
Despite the existence of a garden variety of definitions of continuous dependence for reasons of clarity it is useful to settle with a particular formulation (although we need to apply care to not to over-generalize the particular results we obtain). The archetype case of a mathematical problem used to model a physical process is an initial and/or boundary value problem of a differential equation which expresses a law of a dynamical physical theory47.
47Although there are other relevant and interesting mathematical problems which should also be discussed in the wider context of well posedness – difference equations are among them (seeLax and Richtmyer(1956) or examples e.g. inLavrent’ev and Savel’ev(2006)) – we are going to restrict our attention to this archetype case.
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In the majority of this work we rely on a general and widely used framework developed for abstract differential equations by Peter Lax in the 1950’s. This abstract framework is general enough to handle many fundamental dynamical equations in physics – such as the Maxwell, the Schr¨odinger, the Dirac equation for free particles – and it applies to both initial and initial/boundary value problems (as information about boundary values can be encoded in the domain of the utilized operator A). Appendix A gives a more detailed mathematical background, but the basic ideas can be easily introduced.
Let A be a densely defined operator in an arbitrary Banach space E with a norm k.k.
Consider the equation
u0(t) =Au(t) (−∞< t <∞) (2.18) A solution of (2.18) is a function t → u(t) such that u(t) is continuously differentiable for
−∞< t <∞, u(t) is in the domain D(A) of A, and (2.18) is satisfied for−∞< t <∞.
Definition 4. We say that the Cauchy problem for (2.18) is well posed in the sense of Lax (or simply well posed) in−∞< t <∞ if the following two assumptions hold:
(1) Existence of solutions for sufficiently many initial data: There exists a dense subspace D of E such that, for any u0 ∈D, there exists a solution u(.) of (2.18) in −∞< t <∞ with
u(0) =u0. (2.19)
(2) Continuous dependence of solutions on their initial data: There exists a function C(t) defined for −∞< t <∞ such that C(t) and C(−t) are nondecreasing, nonnegative, and
||u(t)|| ≤C(t)||u(0)|| (−∞< t <∞) (2.20)
for any solution of (2.18).
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Condition (2) can be given an equivalent (but more palpable) formulation as:
(2’) Let {un(.)} be a sequence of solutions of (2.18) with un(0) → 0. Then un(t) → 0 uniformly on compacts of −∞< t <∞.
One can immediately note that the uniqueness condition in Definition4is missing. This is due to the fact that uniqueness is a mathematical consequence of existence 2.19 and continuous dependence 2.20. In a more general setting uniqueness of solution for a studied initial value problem is often established by showing that solutions of suitably taken, tractable initial value problems converge; continuous dependence is then used to infer from convergence of initial data to convergence of solutions.
The relationship between continuous dependence and uniqueness, assuming existence, is even stronger. As one can see from Theorem 6 of Appendix A, if operator A satisfies an additional condition existence and uniqueness of solutions also imply continuous dependence.
This tight connection may tempt someone to try to give a non-circular justification for uniqueness – to try to give a non-circular justification for determinism – via providing an independent justification for continuous dependence. To do this was the original motivation for my research; the problem, as we have seen, lies in the inherently epistemic character of the usual justifications that are offered for continuous dependence.
Continuous dependence is also instrumental for various types of approximation results.
Solutions of well posed problems can often be approximated by solutions of a series of dif-ferent, more easily tractable differential equations; the time variable may also be discretized and the equation itself can often be replaced by a finite difference approximation48. As it is often indispensable for approximation and discretization continuous dependence is also often needed for the applicability of numerical methods and computer simulation.
48For some results see Chapter 5.7 and onwards inFattorini(1983). Note that results about approximation and the link between continuous dependence and uniqueness donot require the norm to be approximately measurable or operationally significant; mathematical results do not require an interpretational link between the formal apparatus and measurement.
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Finally, assuming that we are fine with some pre-established fit, continuous dependence allows us to conveniently choose a relatively simple representative from a set of given data that is compatible with measurement results and to predict by calculating the trajectory of a single, and hopefully simple solution. As given data can only be obtained up to a certain imprecision lack of continuous dependence would necessitate tracing the trajectory of every solution in order to arrive at the set of possible trajectories which may not be fitted within bounds. That we may represent time development of systems using a single solution is a major convenience.
What is the status of well posedness in current research? Even though Hadamard seems to have believed that failure of well posedness would not occur in world obeying physical laws, mathematical research unearthed many examples of problems with physically relevant equations that are not well posed; facing the counterexamples in the middle of the 20th century the reference guide of Courant and Hilbert opined that
Nonlinear phenomena, quantum theory, and the advent of powerful numerical methods have shown that “properly posed” problems are by far not the only ones which appropriately reflect real phenomena. So far, unfortunately, little mathematical progress has been made in the important task of solving or even identifying and formulating such problems that are not “properly posed” but still are important or motivated by realistic situations. (Courant and Hilbert;1962, p. 230)
Subsequent research has emphasized the importance of the mathematical choices we make in formulating our equations; in particular whether an equation is well posed in the sense of Definition4 depends on the choice of the Banach space E. Failure of well posedness may then be linked to the inappropriate mathematical choices, as it is emphasized in the mid-80’s by the author of The Cauchy Problem:
A look at the vast amount of literature produced during the last two decades on deter-ministic treatment of improperly posed problems (including numerical schemes for the computation of solutions) would appear to prove Hadamard’s dictum wrong. However, it may be said to remain true in the sense that, many times, a physical phenomenon appears to be improperly posed not due to its intrinsic character but to unjustified use of the model
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that describes it; for instance the model may have “unphysical ” solutions in addition to the ones representing actual trajectories of the system, bounds and constraints implicit in the phenomenon may be ignored in the model, the initial and boundary value problems imposed on the equation may be incorrect translations of physical requirements, and so on. [...] With the possible exception of the reversed Cauchy problem in Section 6.1, none of the problems and examples in this chapter [which is an overview of ill posed problems in physics – B. Gy.] would probably be recognized by specialists as improperly posed, but rather as properly posed problems whose correct formulation is somewhat nonstandard.
[...] For this reason such non well posed equations are often referred to as weakly ill posed, i.e. they become well posed if we choose another, more suitable space E. (Fattorini;1983, p. 347,375)
In AppendixAwe give examples for physical laws that have a well posed Cauchy problem using one formulation but have a not well posed Cauchy problems using another. We are also going to take this insight as our stepping stone for developing arguments for the necessity of well posedness for physical possibility. In order to proceed first we need to fix the sense of physical possibility that is being invoked in these discussion.
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