Colombeau’s Theory of Generalized Functions

Abstract

The theory of distributions, founded by S. L. Sobolev and L. Schwartz, shows great power and flexibility in its natural domain, the theory of linear partial differential equations. Over the past five decades, numerous publications have contributed to an elaborate solution concept for such equations. As an example we mention the Malgrange-Ehrenpreis theorem showing that any constant coefficient linear PDE possesses a fundamental solution within the space of distributions. However, the inherent limitations of distribution theory, even within the realm of linear PDEs, became apparent as soon as 1957 when H. Lewy ([Lew57]) gave an example of a linear PDE with smooth coefficient functions without solutions in D′. Moreover, its structure as a space of linear functionals does not lend itself to a definition of a “multiplication” of distributions. In fact, various “impossibility results” show that an associative, commutative product on D′ would not coincide with various “natural” products on subspaces of D′. Nevertheless, there are quite a number of instances displaying a need for a concept of multiplication of distributions. Here is a list of some of them:

• 1. Nonlinear PDEs with singular data or coefficients (shock waves in systems from hydrodynamics and elasticity, delta waves in semilinear hyperbolic equations with rough initial data, propagation of acoustic waves in discontinuous media, Schrödinger equations with strongly singular potential, nonlinear stochastic PDEs with white noise excitation, Lie group transformations of generalized functions, ...).

• 2. Intrinsic problems in distribution theory (restriction to submanifolds, calculation of convolutions via Fourier transform, ...).

• 3. Renormalization problems in quantum field theory.

• 4. Singularities in nonlinear field theories, in particular in general relativity (ultrarelativistic limits of spacetime metrics, distributional curvature of cosmic strings, geodesic equations in distributional geometries ...).

• 5. Microlocal regularity and propagation of singularities in nonlinear PDEs or in linear PDEs with non-smooth coefficients.