3 The Initial-Value Problem

We start here with a discussion of hyperbolic evolution problems on the infinite domain n ℝ. 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 [259Jump To The Next Citation Point]. 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.

 3.1 Linear, constant coefficient problems
  3.1.1 Well-posedness
  3.1.2 Extension of solutions
  3.1.3 Algebraic characterization
  3.1.4 First-order systems
  3.1.5 Second-order systems
 3.2 Linear problems with variable coefficients
  3.2.1 The localization principle
  3.2.2 Characteristic speeds and fields
  3.2.3 Energy estimates and finite speed of propagation
 3.3 Quasilinear equations
  3.3.1 The principle of linearization
 3.4 Abstract evolution operators

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