2.2 First Order Systems.2 The Theory of Linear 2 The Theory of Linear

2.1 Existence and Uniqueness of Smooth Solutions. 

Let tex2html_wrap_inline1400 be the space of functions of the form:

displaymath1402

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 (tex2html_wrap_inline1404). What's more, since the Paley-Wiener theorem holds, they are analytic.

To this space also belongs the function,

displaymath1406

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 tex2html_wrap_inline1400 always produces solutions which at each constant time slice are in tex2html_wrap_inline1400 . In fact, defining a tex2html_wrap_inline1400 solutions as:

definition91

a direct application of the uniqueness of the Fourier representation for smooth functions shows:

lemma96

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 tex2html_wrap_inline1400 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.

definition99

Remarks:

theorem106

where the above norm is the usual operator norm on matrices.

If a system is well posed for the tex2html_wrap_inline1464 norm, [recall that the tex2html_wrap_inline1464 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 tex2html_wrap_inline1468 . The above theorem reduces the problem of well posedness to an algebraic one which we further refine in the following theorem:

theorem110

This result is central to the theory. The proof that ii) implies i) is simple and follows directly from the inequality:

displaymath1480

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 tex2html_wrap_inline1364 by Fourier Transform).



2.2 First Order Systems.2 The Theory of Linear 2 The Theory of Linear

image 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