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.

Faculty: Gui-Qiang Chen and Qian Wang

DPhil students: Matthew Schrecker, Matthew Rigby and Chao Yi