Centre for Analysis and Nonlinear Partial Differential Equations
Lead Research Organisation:
University of Edinburgh
Department Name: Sch of Mathematics
Abstract
Nonlinear partial differential equations (PDE) are of universal applicability in the modelling of real-life situations from the flow of air around a wing to the behaviour of financial markets. They are also a natural language for describing the laws of mathematical physics and differential geometry. Their study poses profound intellectual challenges to pure mathematicians as well as important computational problems where accurate numerical data is required in specific applications. Despite its international importance and intense research activity on several fronts, including important breakthroughs in recent years, the UK appears to lag behind its competitors in this area.The present proposal is to establish the Centre for Analysis and Nonlinear Partial Differential Equations, run jointly by the University of Edinburgh and Heriot--Watt University at Edinburgh. This centre will improve the UK's current position through a number of specific actions:--- appointment of outstanding researchers in areas under-represented in the UK--- a programme of instructional workshops open to researchers in the UK and beyond--- two major research workshops on current trends and developments in nonlinear PDE--- a substantial visitor programme to bring the world's best researchers to the UK to give high-profile lectures and establish new research contacts--- development of new research links with industry and other interested parties--- development of new undergraduate and graduate courses in analysis aimed at meeting the needs of the next generation of researchersThe proposal comes from the Maxwell Institute of Mathematics, which is a new joint venture combining the strength of mathematical sciences at the University of Edinburgh and Heriot-Watt University. Funded by the Scottish Funding Council and the Office of Science and Technology, the Maxwell Institute aims to be a pre-eminent centre for research and post-graduate training in the mathematical sciences, offering an environment able to attract and foster the very best mathematical talent from around the world. The Maxwell Institute is one of five joint research initiatives, the others covering a wide range of topics in engineering and geoscience. The present bid will take advantage of the Maxwell Institute's position alongside the other joint research initiatives to develop new collaborations and applications of nonlinear PDE in these areas.The other distinctive feature of this proposal is the presence of the International Centre for Mathematical Sciences (ICMS) which is a joint initiative of the mathematicians at Edinburgh and Heriot-Watt which was set up in 1990. Since then ICMS has developed a reputation for the running of high-level international instructional and research workshops, and the infrastructure it provides will be crucial in organizing the proposed workshops. At the same time, these workshops provide a broadening of ICMS's current activities and will add to its international reputation.The new research grouping will be managed by a Scientific Steering Committee composed of two mathematicians from each of University of Edinburgh and Heriot-Watt University, and also including at least one representative from industry and at least one person from overseas. The committee will be regularly consulted, especially on the workshop and visitor programmes.
Publications
Andersson L
(2013)
A Decay Estimate for a Wave Equation with Trapping and a Complex Potential
in International Mathematics Research Notices
Andersson L
(2014)
Second order symmetry operators
in Classical and Quantum Gravity
Andersson L
(2016)
Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior
in Journal of Hyperbolic Differential Equations
Anthony Carbery (Author)
(2011)
GLOBAL EXISTENCE FOR AN L-2 CRITICAL NONLINEAR DIRAC EQUATION IN ONE DIMENSION
in Advances in differential equations
Anthony Carbery (Author)
(2010)
Elliptic equations in the plane satisfying a Carleson measure condition
in Revista matematica iberoamericana
Anthony Carbery (Author)
(2012)
LOCAL AND GLOBAL WELL POSEDNESS FOR THE CHERN-SIMONS-DIRAC SYSTEM IN ONE DIMENSION
in Differential and integral equations
Boulton L
(2012)
On the Stability of a Forward-Backward Heat Equation
in Integral Equations and Operator Theory
Bournaveas N
(2011)
Local well-posedness for the space-time Monopole equation in Lorenz gauge
in Nonlinear Differential Equations and Applications NoDEA
Carbery A.
(2014)
The integrability of negative powers of the solution of the Saint Venant problem
in Annali della Scuola normale superiore di Pisa - Classe di scienze
| Description | Making the appointments took longer than anticipated: we had to run recruitment exercises three years running. On each occasion our advertisements attracted a strong field of applicants, but in the first two years, our top choices preferred appointments elsewhere---on more than one occasion we were in direct competition with high-profile universities in USA/Canada. In particular the second lecturer appointment started at Edinburgh only in September 2009 and this initially slowed the research development of the centre. The problem of regularity of incompressible minimisers of stored energy functionals is one of the central open questions in nonlinear elasticity. In a recent paper Karakhanyan has linked the regularity of minima to a Monge-Ampere type equation in two spatial dimension and proved the half Holder continuity of weak solutions. This is a novel and promising approach that can be applied to larger class of problems with hard Jacobian constraints. With Wang, he has established several new results on reflector antenna design problems. They have introduced and successfully applied the method of stretch functions and have also simulated extensive research around non-variational problems involving Monge-Ampere type equations. They have explored the possibility of constructing monotonicity formulas for free boundary problems driven by nonlinear elliptic operators (e.g. p-Laplacian), and have introduced a discrete version of Caffarelli's monotonicity formula in two spatial dimension. This is joint work with DiPierro and Shahgholian. Finally, in homogenisation of PDEs some advances have been made in the following two directions: (a) Homogenization of thin obstacle problem for p-Laplacian (jointly with Stromqvist) where they introduce a novel method of proving compactness via quasi-uniform convergence and p-capacity estimates and (b) settled the conjecture of De Giorgi about periodic oscillations of ode systems (with Shahgholian). Differentiation along the generators of classical symmetries played a crucial role in the proof of the stability of empty space in Einstein's theory of relativity. The rotating Kerr family of black holes, which are the most physically relevant, have far fewer classical symmetries arising from Killing vectors, but have a Killing tensor that has been known since the 1970s. Blue and Andersson were able to use this to prove decay for the wave equation outside a Kerr black hole. This preprint has already been widely cited. Although the effect of trapping geometries, which obstruct the dispersion of waves, and of complex potentials, which break the variational structure and conservation laws for wave equations, had been previously understood, their interaction introduces additional complexities. Blue and co-authors have analysed scalar wave equations with these features in the analysis of both the Maxwell and linearised Einstein equations outside a Kerr black hole. Although the Kerr black holes had been known both to possess a Kiling tensor and to have a differential operator that commutes with the Maxwell equation, the relation between these was not as clear as in the case of the wave equation. They were able to identify a simple condition under which the tensor generates a commuting operator. Detailed case studies can be submitted upon request. The S&I award of the CANPDE to the Maxwell Institute has firmly cemented in place the role of Edinburgh as a leading location internationally in analysis and nonlinear PDEs. Analysis and PDEs have gone from strength to strength in both the University of Edinburgh and Heriot-Watt where, in addition to those (UoE) detailed in this document, several more appointments have been made (including Coutand, Beck, Banjai, Lindsay, Staubach). This puts Analysis and PDEs right at the centre of the Maxwell Institute's research agenda which is fully in line with that of the two universities. The universities are enthusiastically supporting the recently announced CDT in Analysis and its Applications and it is envisaged that several further appointments in this area will be made in order to further expand activity in analysis and PDEs. |
| Exploitation Route | We have made three outstanding permanent appointments at lecturer level (2 at UoE) with expertise in theoretical PDE with applications in many areas including continuum mechanics and mathematical physics. They have joined the Edinburgh analysis group which includes two researchers working in PDE (Bournaveas and Dindos), world-leading harmonic analysts (Carbery and Wright) and Gyongy, who is a pre-eminent researcher at the interface of PDE, stochastic analysis, and numerics. We have also recruited a Chancellor's Fellow, Tadahiro Oh, who works in the area of probabilistic dispersive nonlinear PDE. Oh has recently been awarded an EPSRC Early Career Fellowship. Both of our permanent lecturers (Blue and Karakhanyan) were awarded EPSRC First Grants. Their postdocs (Backdahl and Dipierro) are jointly funded by CANPDE. CANPDE is also jointly funding one of the School's Whittaker Research Fellows (Choffrut). We have trained the following PhD students funded by CANPDE Candy (nonlinear waves), supervised by Bournaveas; Hall, (PDE/harmonic analysis), supervised by Gyongy Plus the following were supported by other funding sources Aleksanyan (Karakhanyan); Alhojilan (Gyongy); Alnafisah (Gyongy); Bakas (Wright); Dareiotis (Gyongy); Gerencser (Gyongy); Hickman (Wright); Iliopoulou (Carbery); Kirsch (Dindos); Kumar (Sabanis); Leahy (Gyongy); Ovcharov (Bournaveas); Rodrigues (Rasonyi); Villarreal (Rasonyi); Vitturi (Wright); Zhang (Szpruch). In 2014 we were awarded a Centre for Doctoral Training in Analysis and its Applications. This will allow us to continue training substantial numbers of PhD students in the areas covered by CANPDE. David Rule was a valuable addition to the group as post-doc for 2 years. In order to bolster interest in analysis and nonlinear PDE across the country, CANPDE has continued to organise mini-symposia (19, www.maths.ed.ac.uk/~pblue/PDEminisymp/) and crash courses (8). We support workers in PDE, especially early career researchers, from across the UK in all of our activities; this included supporting the Young Applied Analysts in the UK conference held in Glasgow and the 2nd Scottish PDE Colloquium at ICMS. Other contributions to the UK's research capability include a national PDE email list and events diary (jointly run with OXPDE). Sustainability: the main institutional commitment is to take over the salary costs of Blue (as of December 2013) and Karakhanyan (as of the end of the grant). Choffrut will also be funded by UoE for a further year. The remaining funding for Backdahl and Dipierro will come from the first grants of Blue and Karakhanyan. The appointment of Oh and other expected appointments in the area point to the long term commitment from the two universities to the sustainability of research in analysis and nonlinear PDEs in the Maxwell Institute. |
| Sectors | Other |
| Description | We have worked closely with OXPDE on several joint ventures including `Ball 60', a workshop on Geometric Analysis, Elasticity, and PDE held at Heriot Watt University in 2008 and Karakhanyan was one of the organisers of the jointly organised International Conference on Nonlinear Partial Differential Equations in 2012 in Oxford. Karakhanyan is also co-organising a PDE conference to be held in Sussex in September 2014. We have developed collaborations within the UK: Jon Bevan & Bin Cheng (Surrey), Ali Taheri (Sussex), Mike Cullen (Met Office); and internationally: L. Caffarelli (Austin), M. Feldman (Wisconsin Madison), H. Shahgholian, E.Lindgren and M.Stromqvist (KTH Stockholm), D. Danielli (Purdue), J-F. Rodrigues (Lisboa), J-P Nicolas (Brest) and D Hafner (Grenoble). |
| Impact Types | Cultural |