Nonlinear PDEs research group

Members of this research group include: E. Varvaruca, B. Pelloni plus research students.

Current research being undertaken includes

The rigorous mathematical analysis of free-boundary problems for nonlinear partial differential equations (PDE)
E. Varvaruca and research students

In such problems, the domain in which one is solving a partial differential equation is not given, but has to be found as part of the solution of the problem. Research involves studying the existence and the geometric properties of free boundaries, their smoothness and the nature of their singularities. Free-boundary problems are ubiquitous in fluid dynamics, arising for example in the description of free-surface and interfacial waves.

Large-amplitude steady gravity water waves with vorticity are a particular focus of work. Methods of global bifurcation theory to prove the existence of new types of such waves are being used, and new a priori estimates derived for the amplitude and the fluid velocity of such waves by using elliptic PDE techniques such as the maximum principle.

Boundary value problem for integrable nonlinear equations
B. Pelloni plus research students (with T. Fokas, Cambridge)

I am interested in the solvability and solution representation of boundary value problems posed ni a variety of domains for nonlinear PDEs in the integrable class. I have worked extensively on developing and making rigorous the integral transform approach of Fokas. This complex-analytic approach is a substantial generalisation of the inverse scattering transform. Information about the unified transform method for boundary value problems can be found here.

Analysis of PDE models of large scale atmospheric flow
B. Pelloni plus research students (with M. Cullen, MET office)

I am currently working, with my PhD student David K Gilbert, on the rigorous analysis of the solution, in physical variables, of the semi-geostrophic equations in 3D. This work is based on results in optimal transport theory and recent estimates of solutions of gradient flows and Hamiltonian ODEs.

Linear PDEs and spectral theory
B. Pelloni plus research students

The Fokas transform approach can be used for the study of linear problems. This study yielded interesting and unexpected results, such as a characterisation theorem on the boundary conditions that yield two-point boundary value problems which are well posed. It also provided a new way to characterise problems whose solution admits a series representation, by exploiting general results in the theory of entire functions of finite exponential sum type. This work has been extended and set on a fully rigorous footing in the PhD research of my student David A. Smith (PhD expected 2011).

 

Image of the Great Wave by Hokusai, ©Trustees of the British Museum