We start here with a discussion of hyperbolic evolution problems on the infinite domain . This is
usually the situation one encounters in the mathematical description of isolated systems, where some
strong field phenomena take place “near the origin” and generates waves, which are emitted
toward “infinity”. Therefore, the goal of this section is to analyze the well-posedness of the
Cauchy problem for quasilinear hyperbolic evolution equations without boundaries. The case
with boundaries is the subject of Section 5. As mentioned in the introduction (Section 1),
the well-posedness results are fundamental in the sense that they give existence (at least local
in time if the problem is nonlinear) and uniqueness of solutions and show that these depend
continuously on the initial data. Of course, how the solution actually appears in detail needs to
be established by more sophisticated mathematical tools or by numerical experiments, but it
is clear that it does not make sense to speak about “the solution” if the problem is not well
posed.
Our presentation starts with the simplest case of linear constant coefficient problems in Section 3.1, where solutions can be constructed explicitly using Fourier transform. Then, we consider in Section 3.2 linear problems with variable coefficients, which we reduce to the constant coefficient case using the localization principle. Next, in Section 3.3, we treat first-order quasilinear equations, which we reduce to the previous case by the principle of linearization. Finally, in Section 3.4 we summarize some basic results about abstract evolution operators, which give the general framework for treating evolution problems including not only those described by local partial differential operators, but also more general ones.
Much of the material from the first three subsections is taken from the book by Kreiss and Lorenz [259].
However, our summary also includes recent results concerning second-order equations, examples of wave
systems on curved spacetimes, and a very brief review of semigroup theory.
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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