Overview

The main tool to discuss constant-coefficient problems (with initial data given on the whole space) is the Fourier transformation. It allows us to decompose general initial data into pure exponentials. One easily observes that in case of constant coefficients the time evolution of each pure (spatial) exponential can be treated separately. We introduce the symbol P( i ?) of a differential operator P( ?/ ?x) and obtain

as the solution for initial data . Well-posedness of the Cauchy problem can be characterized in terms of estimates for the symbol.

We will start out with some special cases, namely hyperbolic and parabolic systems in one space dimension. For these, the conditions for the symbol are easily checked, and one can solve the Cauchy problem. In Section 2.3 we characterize families of matrices A for which the exponential e ^At , t ?0, is uniformly bounded. This result, which is central in a general theory of well-posedness, will be applied in Section 2.4 to characterize those first-order systems (in any number of space dimensions) which lead to well-posed Cauchy problems; the corresponding systems are called strongly hyperbolic. An important example is given, the compressible Euler equations linearized about a constant flow. Similarly, linearization about a constant flow of the viscous compressible Navier-Stokes equations leads to a mixed hyperbolic-parabolic system; applying the same general principles, we obtain well-posedness of the Cauchy problem and can solve these linearized problems by Fourier transformation.

Though constant-coefficient equations...