A Banach Space of Analytic Functions For Constant Coefficient Equations of Evolution

Extract

The spaces involved in the theory of equations of evolution (that is, the theory of semi-groups) are such that the infinitesimal generators are only densely defined. For the infinitesimal generator to be everywhere defined and smooth (that is, differentiable), one must work with a Frechet space. This is especially important in the non linear theory of Moser (see Moser [3] and Marsden [2]). If the spaces were Banach spaces, the theory would reduce to the classical Picard theory for ordinary differential equations. See Lang [1], for a discussion of the classical theory. In fact in this case the existence theory is both simpler and more comprehensive, because we obtain the important fact that the flow is a diffeomorphism (the solutions depend smoothly, and not merely continuously on the initial data). This has other advantages too, since the theory for smooth flows on Banach manifolds is well developed. (For example, see Marsden [2], theorem. 6.10 for an application we have in mind.)