Supervisor: Dr Arick Shao
A wide variety of phenomena in science, economics, and engineering are mathematically modelled by partial differential equations, or PDEs. This project aims to study asymptotic and control properties of special classes of PDEs that are “critically singular”, in that they contain singular or degenerate coefficients that cannot be treated using perturbative methods. Such equations not only appear in many areas of physics (e.g. in gravitation, shock formation, and diffusion), but are also of mathematical interest, as their solutions often exhibit novel behaviours that are not yet well understood.
One branch of this project deals with the control theory of PDEs, which investigates whether solutions of a PDE can be steered to a given desired state using limited data. Here, our objective is to study whether solutions of various linear and nonlinear (geometric) wave or heat equations on a bounded domain, with potentials that become critically singular at the boundary, can be controlled by appropriately imposing its boundary data.
Such critically singular wave equations are closely connected to problems in relativity, gravitation, and holography, in particular in the AdS/CFT correspondence in theoretical physics. Similarly, singular heat equations can be related, for instance, to fluids with degenerating viscosity. Although controllability of PDEs is a well studied problem, very little is currently known for the above-mentioned critically singular PDE, especially in higher dimensions.
A second branch of this project studies the dynamical and asymptotic properties of linear and nonlinear wave equations and hyperbolic systems that become “weakly hyperbolic”, i.e. in which the hyperbolicity of the PDE degenerates. When the lower-order coefficients of such PDE degenerate at a “critical” rate with respect to the weak hyperbolicity, their solutions exhibit a rich set of behaviours, including fractional losses of regularity, that are still not fully understood.
Such phenomena can in fact be directly connected to PDEs with coefficients that become critically singular in time. Interesting examples of such equations can be found in a variety of physical settings, for instances at big bang singularities or during shock formation. Thus, a key aim of this programme is to develop a precise quantitative understanding of asymptotic behaviours of such linear and nonlinear PDEs at the singular or weakly hyperbolic time.
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