Since the integral exists at each point, it can be
differentiated inside the integral sign where it gives another
compactly supported integrand, thus these functions are smooth (). What's more, since the Paley-Wiener theorem holds, they are
analytic.
To this space also belongs the function,
This function is smooth not only along space directions, but
also along time directions. In fact, it is straightforward to
check that the function satisfies the evolution equation above
for the initial condition
f
(x). Thus we see that initial data in
always produces solutions which at each constant time slice are
in
. In fact, defining a
solutions as:
a direct application of the uniqueness of the Fourier representation for smooth functions shows:
Thus we see that there are plenty of smooth solutions,
whatever the system is. But it was realized by Hadamard, [24], that there were not enough solutions, since the space
is not closed. Furthermore, in general there are no topologies
on the space of initial data, and of solutions for which
solutions depend continuously on initial data. Lack of continuity
of solutions with respect to their initial data would not only
imply lack of predictability from the physical standpoint, for
all data are subject to measurement errors, but also lack of
realistic possibilities of numerically computing solutions, due
to truncation errors. Thus it is important to characterize the
set of equations for which continuity holds. There are several
possibilities for the choice of the topologies for the spaces of
initial data and of solutions. Here we restrict consideration to
those which have been more prolific with respect to results and
generalizations to non-linear, non-constant coefficient equations
systems.
Remarks:
where the above norm is the usual operator norm on matrices.
If a system is well posed for the
norm, [recall that the
norm of a function is the square root of the integral of its
square], then it is well posed for any other Sobolev norm, (as
follows from the above theorem), since the constants are
independent of
. The above theorem reduces the problem of well posedness to an
algebraic one which we further refine in the following
theorem:
This result is central to the theory. The proof that ii) implies i) is simple and follows directly from the inequality:
that is, from the construction of an
energy norm
. We see that for any well posed problems this special energy
norm can be constructed, so one can always attempt to approach
the problem by trying to find, usually with the help of the
physics behind the problem, the correct energy norm. Condition
ii)
is usually referred to as the
semiboundedness of the operator
P(D) with respect to the norm
H
(induced on functions in
by Fourier Transform).
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Hyperbolic methods for Einstein's Equations
Oscar A. Reula http://www.livingreviews.org/lrr-1998-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |