Chapter 1 - The Cauchy Problem for Some Equations of Mathematical Physics: The Abstract Cauchy Problem

Summary

The purpose of this chapter is to introduce the notions of properly posed Cauchy problem in t ≥ 0 (Section 1.2) and in – ∞ < t < ∞ (Section 1.5) for the equation u′(t) = Au(t) in an arbitrary Banach space E. These definitions will be fundamental in the rest of this work. In the case A is a differential operator in a function space E, there are relations with Cauchy problems that are well posed in the sense of Hadamard; these relations are explored in Section 1.7. The rest of the chapter is understood as motivation for the central idea of well posed Cauchy problem: several equations of mathematical physics are examined by ad hoc Fourier series and Fourier integral methods with the aim of discovering properties of solutions which will be generalized later to wide classes of equations.

Sections 1.1 and 1.3 deal with the heat-diffusion equation in a two-dimensional square. We find that (a) the equation produces a properly posed Cauchy problem in the spaces L^p (1 ≤ p < ∞) and in spaces of continuous functions; (b) nonnegative initial data give rise to nonnegative solutions; (c) the L¹ norm of nonnegative solutions is preserved in time; (d) solutions become extremely smooth in arbitrarily small time. All of these properties are instances of general theorems on parabolic equations to be found in Chapter 4.

Section 1.4 treats the Schrödinger equation. […]