GAnG - Dr Arick Shao
Control and Asymptotics of Critically Singular Partial Differential Equations
Supervisor: Dr Arick Shao
Project description:
A wide variety of phenomena in science and engineering are mathematically modelled by partial differential equations (PDEs). This project aims to study asymptotic and control properties of special classes of PDEs that are critically singular, in that the equations contain coefficients that become infinite at a rate that dramatically affects the behaviour of solutions. Such equations appear in many important models of physical interest, for instance gravitation, fluids, and diffusion. These equations are also of independent mathematical interest, as they are known to be especially difficult to treat, and their solutions exhibit novel behaviours that are not yet well understood.
One branch of this project deals with the control theory of PDEs, which studies whether solutions, representing some physical phenoma, can be actively steered over time to a desired state using limited controls. Here, our objective is to study whether solutions of various linear and nonlinear wave and heat equations on a bounded domain, with potentials that become critically singular at the boundary, can be controlled by appropriately imposing its boundary conditions. Although control of PDEs in general is a well studied problem, very little is currently known for the above-mentioned critically singular PDE, especially in higher dimensions.
Such critically singular wave equations are closely connected to problems within relativity, gravitation, and holography, in particular in the AdS/CFT correspondence (which has played an influential role in recent efforts to unify relativity and quantum mechanics). Consequently, results for these equations would lead to immediate applications toward these physical models. Similarly, singular heat equations can, for instance, model fluids with degenerating viscosity at the boundary.
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, in which the hyperbolic, or roughly “wave-like”, character of the PDE degenerates at a given time. When the lower-order coefficients of such PDE degenerate at a critical rate with respect to the hyperbolicity, their solutions exhibit a rich set of unusual behaviours that are not fully understood – one striking example is loss of regularity, in which solutions become less smooth at the degenerate time.
Such phenomena can in fact be directly connected to PDEs with coefficients that become critically singular in time. One aim in this research is to determine the precise asymptotic behaviour of solutions at this degenerate time. Another goal is the the converse problem of scattering – whether one can find solutions arising from appropriately prescribed data at the degenerate time.
Interesting examples of such PDEs can be found in a variety of physical settings. In relativity and cosmology, the degenerate time could model a big bang singularity, while for fluids, such a degenerate time may capture shock formation. Therefore, a precise description of the asymptotics of these weakly hyperbolic PDEs would lead to better understanding of a variety of physical models, for instance a more precise description of behaviour at a big bang singularity.
Funding Notes:
This project is open to candidates applying for EPSRC/Underrepresented Studentships.
Further information:
How to apply
Entry requirements
Fees and funding
PhD Information Session 2026:
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