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.