HARMONIC ANAL AND PDE · Harmonic analysis techniques for partial differential equations in mathematical physics and geometry

Линиите свързват координатора с партньорите.

Накратко на български

Цел на проекта

We plan to study elliptic and parabolic non-linear partial differential equations that arise in geometry and mathematical physics on rough domains. By rough we mean Lipschitz domains, however in some cases we will consider general singularities (such as cusps or cracks). Problems on rough domains occur naturally in geometry, physics and other disciplines. Consider for example a classical problem in geometry; the 2d conformal change of metric. Given a metric on a domain we look for a new metric conformal with the original one of prescribed (negative) curvature (typically -1).This problem leads to a differential equation, which can be considered with various boundary conditions. Of particular interest is the condition that the solution blows up uniformly at the boundary. If such a solution exists, then the domain in the new metric can be geodesically complete. This problem is interesting on any domain as it leads to a classification of two-dimensional Riemann surfaces. It turns out that harmonic analysis provide s many tools to successfully tackle this and similar problems.Other examples of problems that naturally arise on non-smooth domains include the problem of modelling cracks in elasticity or a fluid flow on domains with corners etc. Our goal is to contribute to the development of new techniques to handle these problems. One particular problem we plan to study is the stationary Navier-Stokes equation. The time dependent Navier-Stokes equation is one of the most studied problems since it governs the motion of a viscous fluid. Even though the stationary equation is more understood, many problems remain open.One of them is the global existence of solutions for arbitrary large data in any dimension. Even a partial result in this direction will contribute to advancement in other areas of mathematics; such as fluid dynamics. For this reason we also plan to focus on a related equation - the quasi-geostrophic equation arising in meteorology.

Оригинален текст от CORDIS (на английски).

Участници

• UNIVERSITY OF EDINBURGH · EDINBURGHКоординаторОбединеното кралство

Връзки

• Виж в CORDIS