Research areas

The centre focuses on the fundamental analysis of PDEs, and numerical algorithms for their solution, together with specific PDEs arising in wide-ranging areas. Research interests include:

Nonlinear Hyperbolic Systems

The programme is toward mathematical research on nonlinear hyperbolic systems of conservation laws and related nonlinear problems, along with the analysis and development of efficient nonlinear ideas, techniques and approaches.

Such nonlinear PDE especially include the Euler equations for compressible fluid flows. The current focus is on three interrelated aspects:

(i) well-posedness: existence, uniqueness, and stability;
(ii) analysis of entropy solutions: compactness, structure, regularity, and asymptotic behaviour; and
(iii) stability and interactions of nonlinear waves including shock waves, rarefaction waves, vortex sheets, and entropy waves.

The objective of this research is twofold:

(a) investigate important nonlinear hyperbolic systems of conservation laws and related problems to gain new physical insights, to guide the formulation of efficient nonlinear techniques, and to find correct mathematical frameworks in which to pose the nonlinear hyperbolic systems of conservation laws and develop the approximate/numerical methods that converge stably and rapidly;

(b) analyze and develop nonlinear techniques to formulate new, more efficient nonlinear ideas, techniques and approaches and to solve various more important nonlinear hyperbolic systems of conservation laws arising from mathematics and the other sciences. Related nonlinear partial differential equations of mixed hyperbolic-elliptic type in mechanics and geometry are also analysed.

Connections between hyperbolic conservation laws/shock waves and nonlinear PDE in relativity are also explored.

Please contact Gui-Qiang Chen for more details.

Post-doctoral Researchers in this research area include Laura Caravenna and recently Wei Xiang who joined the Centre in September, 2012.

DPhil students in this research area include Ben Stevens.

Fluid Mechanics

Research activities in Fluid Mechanics cover a wide range of aspects in the mathematical theory of the classical Euler and Navier–Stokes equations describing the flow of incompressible fluids and more recent models in Non-Newtonian Mechanics including generalized Newtonian fluids, visco-elastic and visco-plastic fluids. Among those aspects are: well-posedness of the corresponding boundary-value and initial boundary-value problems, qualitative properties of solutions, developments of fine analytic methods (like unique continuation, Liouville’s theorems, maximum principles, Harnack’s inequalities, etc) for resolving challenging problems inspired by the fundamental mathematical questions in Fluid Mechanics.

Please contact Gregory Seregin for more details.

Liquid Crystals

Please contact John Ball for more details.

Materials Science

Please contact John Ball for more details.

Numerical Analysis of Nonlinear PDE

Please contact Endre Suli for more details.

Relativity

Mathematical general relativity is concerned with all mathematical aspects of Einstein’s theory of gravitation. This includes the study of general relativistic initial data set, of stationary black holes (both these topics typically lead to problems in the theory of nonlinear elliptic partial differential equations), as well as problems in the asymptotic behavior of the gravitational field in the radiation regime, or near singularities, and stability questions (leading to hyperbolic PDEs problems).

Please contact Paul Tod for more details. 

Solid Mechanics

The Centre worked closely with OxMOS, the New Frontiers in the Mathematics of Solids research programme, and with the newly formed Oxford Solid Mechanics initiative.