Herbert Koch: The Korteweg–de Vries hierarchy at low regularity

Abstract:

In this talk I report on joint work with Baoping Liu and Friedrich Klaus. We prove existence and uniqueness of weak solutions to all equations in the Gardner hierarchy for initial data in \(L^2(R)\). As a consequence we obtain uniqueness of weak solutions to all equations of the Korteweg–de Vries hierarchy for \(L^2(R)\) initial data and a continuous extension of the flows to \(H^1\).

The proof makes heavy use of the Miura map and of the commuting vector field method of Killip and Visan.