Theory of Distributions and Linear Differential Equations

Ruoyu Wang (University College London)

The theory of distributions, also named the theory of generalised functions, was well developed by Laurent Schwartz in 1940s. The theory is a generalisation of classical analysis, which provides a wider framework in which we can reformulate and develop classical problems in a perspicuous language. Its main idea is to view functions as continuous linear functionals acting on the space of 'well-behaved' functions. In this sense of extension, we can define some functions which are irregular, for example the Dirac delta function, and even derivatives at discontinuities. This allows us to find a reasonable explanation to non-classical solutions to PDEs, which are not smooth enough in the classical sense, and moreover provides a general approach to solve linear PDEs.

The talk will quickly go through the following points: i) definitions and basic operations; ii) irregular distributions; iii) derivatives at discontinuities; iv) examples of solving linear ODEs and PDEs by the theory; v) short comments on weak solutions to the Navier-Stokes equations.