We consider an abstract initial value problem
(29) |
To this end, let us first review a few ways of defining the exponential of a matrix.
1. The power series:
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2. As a limit:
(31) |
3. By diagonalization or, more generally, transforming to Jordan canonical form. This is the procedure we followed in the preceding sections.
4. By Laplace transforms: Consider the equation
and take the Laplace transform: We find the transformed equation which leads to The inversion formula for the Laplace transform then yields where must be chosen larger than the real part of any eigenvalue of A. This leads us to the definition(32) |
For unbounded operators, the power series definition is not useful. Consider the example of the heat equation discussed in the last section A=d^{2}/dx^{2}. Then formally, we would have
(33) |
The definition as a limit seems to suffer from the same defect. However, we can make the following modification to it:
(34) |
Theorem 3
Assume that A is a linear operator defined on a dense subspace D(A) of a Hilbert space H. Assume further that there are constants M and such that exists (as an operator from H to D(A)) for and
(35) |
(36) |
(37) |
The set of operators where is referred to as a semigroup of operators; it is closed under multiplication since . It is in general not possible to extend t to negative values; for instance the heat equation is well posed only for solutions forward in time, not for solutions backward in time.
We note that
(38) |
(39) |
Example 7
To give a simple example, we show how the Hille-Yosida theorem applies to the heat equation, discussed in the previous section. We use the Fourier series representation
(40) |
(41) |
(42) |
(43) |
and consequently,
(44) |
In a somewhat simpler fashion, we could argue that
(45) |
(46) |
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(48) |
(49) |
(50) |
The second argument given in the last example clearly yields a weaker conclusion than the first. On the other hand, it required no knowledge of eigenvalues and eigenfunctions. In general, the representation of in terms of eigenvalues and eigenfunctions is very useful for closed form solutions, if such a representation is available. As a foundation for a general theory, however, such a definition would be too restrictive, since little is known (except for special cases such as self-adjoint operators) about spectral representations. Indeed, there are (physically relevant) examples of operators which do not even have a spectrum, so any attempt to ``diagonalize" is doomed from the start.
Example 8
Consider the equation u_{t}+u_{x}=0, with boundary condition u(0,t)=0 on the interval (0,1). Abstractly, we associate with this the operator Au=-u' with domain . The initial value problem has a well-defined solution, namely, we have
(51) |
(52) |
(53) |
If, however, we consider the problem , i.e.
(54) |
(55) |
In the earlier sections on stability of systems of ODEs, we established a connection between stability and eigenvalues. For infinite-dimensional systems, the issues are more complicated. First of all, there is a more general notion of spectrum. We say that is in the resolvent set of A if the ``resolvent" exists as a bounded operator defined on all of H, and we say is in the spectrum if it is not in the resolvent set. In the finite-dimensional case, the spectrum consists precisely of the eigenvalues, but in infinite dimensions this is no longer the case; for instance, there may be what is known as a ``continuous spectrum."
Example 9
Consider, for instance, the operator
Au[x]=xu[x] | (56) |
(57) |
For any operator which satisfies the hypotheses of the Hille-Yosida theorem, we can define the following quantities:
We call r(A) the spectral bound, and the type of the semigroup. The quantity measures the rate of exponential growth or decay of . The connection between spectrum and stability is established if . While it can be shown that , the converse inequality is in general not true. There are a number of known counterexamples in the literature; a very natural one [13] is the equation
u_{tt}=u_{xx}+u_{yy}+e^{ix}u_{y}, | (58) |
The Laplace inversion formula for (see above) can be exploited to establish a weaker result. Namely, it turns out that
We note that, if there are no spectral values for , i.e. if s>r(A), then the resolvent exists in that half plane. But this does not necessarily imply that it is uniformly bounded. In the finite-dimensional case, this is no issue, since .
For a wide class of problems, including those of Newtonian fluid mechanics, it is known that . However, these results do not include viscoelastic fluids, where the problem is in general still open. Hence we do not know that the onset of instabilities is always associated with eigenvalues crossing the imaginary axis. Nevertheless, many of the instabilities observed clearly show the type of dynamics which is expected from such a scenario, and we shall analyze them in the appropriate framework.