Analysis of Nonlinear Partial Differential Equations
Lead Research Organisation:
UNIVERSITY OF OXFORD
Department Name: Mathematical Institute
Abstract
Partial differential equations (PDEs) are equations that relate the partial derivatives, usually with respect to space and time coordinates, of unknown quantities. They are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of phenomena in the physical, natural and social sciences, often arising from fundamental conservation laws such as for mass, momentum and energy. Significant application areas include geophysics, the bio-sciences, engineering, materials science, physics and chemistry, economics and finance. Length-scales of natural phenomena modelled by PDEs range from sub-atomic to astronomical, and time-scales may range from nanoseconds to millennia. The behaviour of every material object can be modelled either by PDEs, usually at various different length- or time-scales, or by other equations for which similar techniques of analysis and computation apply. A striking example of such an object is Planet Earth itself.Linear PDEs are ones for which linear combinations of solutions are also solutions. For example, the linear wave equation models electromagnetic waves, which can be decomposed into sums of elementary waves of different frequencies, each of these elementary waves also being solutions. However, most of the PDEs that accurately model nature are nonlinear and, in general, there is no way of writing their solutions explicitly. Indeed, whether the equations have solutions, what their properties are, and how they may be computed numerically are difficult questions that can be approached only by methods of mathematical analysis. These involve, among other things, precisely specifying what is meant by a solution and the classes of functions in which solutions are sought, and establishing ways in which approximate solutions can be constructed which can be rigorously shown to converge to actual solutions. The analysis of nonlinear PDEs is thus a crucial ingredient in the understanding of the world about us.As recognized by the recent International Review of Mathematics, the analysis of nonlinear PDEs is an area of mathematics in which the UK, despite some notable experts, lags significantly behind our scientific competitors, both in quantity and overall quality. This has a serious detrimental effect on mathematics as a whole, on the scientific and other disciplines which depend on an understanding of PDEs, and on the knowledge-based economy, which in particular makes increasing use of simulations of PDEs instead of more costly or impractical alternatives such as laboratory testing.The proposal responds to the national need in this crucial research area through the formation of a forward-looking world-class research centre in Oxford, in order to provide a sharper focus for fundamental research in the field in the UK and raise the potential of its successful and durable impact within and outside mathematics. The centre will involve the whole UK research community having interests in nonlinear PDEs, for example through the formation of a national steering committee that will organize nationwide activities such as conferences and workshops.Oxford is an ideal location for such a research centre on account of an existing nucleus of high quality researchers in the field, and very strong research groups both in related areas of mathematics and across the range of disciplines that depend on the understanding of nonlinear PDEs. In addition, two-way knowledge transfer with industry will be achieved using the expertise and facilities of the internationally renowned mathematical modelling group based in OCIAM which, through successful Study Groups with Industry, has a track-record of forging strong links to numerous branches of science, industry, engineering and commerce. The university is committed to the formation of the centre and will provide a significant financial contribution, in particular upgrading one of the EPSRC-funded lectureships to a Chair
Publications
Chen G
(2015)
Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations
in Archive for Rational Mechanics and Analysis
Chen G
(2009)
Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding
in Proceedings of the American Mathematical Society
Chen G
(2012)
Kolmogorov's Theory of Turbulence and Inviscid Limit of the Navier-Stokes Equations in $${\mathbb {R}^3}$$
in Communications in Mathematical Physics
CHEN G
(2018)
Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow
in Acta Mathematica Scientia
Chen G
(2017)
Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems
in Science China Mathematics
Chen G
(2013)
Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows
in Journal of Mathematical Physics
Chen G
(2020)
Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow
in SIAM Journal on Mathematical Analysis
Chen G
(2012)
Global Steady Subsonic Flows through Infinitely Long Nozzles for the Full Euler Equations
in SIAM Journal on Mathematical Analysis
Chen G
(2011)
Nonlinear Conservation Laws and Applications
Chen G
(2011)
On Nonlinear Stochastic Balance Laws
Chen G
(2012)
On Nonlinear Stochastic Balance Laws
in Archive for Rational Mechanics and Analysis
Chen GG
(2018)
Global Weak Rigidity of the Gauss-Codazzi-Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity.
in Journal of geometric analysis
Chen GQ
(2015)
Free boundary problems in shock reflection/diffraction and related transonic flow problems.
in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Choi K
(2014)
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations
in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Choquet-Bruhat Y
(2011)
The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions
in Annales Henri Poincaré
Choquet-Bruhat Y
(2010)
The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions
Chrusciel P
(2011)
Ghost points in inverse scattering constructions of stationary Einstein metrics
in General Relativity and Gravitation
Chrusciel P
(2010)
A uniqueness theorem for degenerate Kerr-Newman black holes
Chrusciel P
(2010)
A Uniqueness Theorem for Degenerate Kerr-Newman Black Holes
in Annales Henri Poincaré
Chrusciel P
(2008)
On Mason's Rigidity Theorem
in Communications in Mathematical Physics
Chrusciel P
(2009)
On the global structure of the Pomeransky-Senkov black holes
Chrusciel P
(2010)
On smoothness of black saturns
in Journal of High Energy Physics
Chrusciel P
(2009)
On higher dimensional black holes with Abelian isometry group
in Journal of Mathematical Physics
Chrusciel P
(2008)
Singular Yamabe Metrics and Initial Data with Exactly Kottler-Schwarzschild-de Sitter Ends
in Annales Henri Poincaré
Chrusciel P
(2011)
A lower bound for the mass of axisymmetric connected black hole data sets
in Classical and Quantum Gravity
Chrusciel P
(2010)
On smoothness of Black Saturns
Chrusciel P
(2009)
Topological Censorship for Kaluza-Klein Space-Times
in Annales Henri Poincaré
Chrúsciel P
(2010)
On the global structure of the Pomeransky-Senkov black holes
in Advances in Theoretical and Mathematical Physics
Condette N
(2010)
Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth
in Mathematics of Computation
Dai S
(2010)
Crossover in coarsening rates for the monopole approximation of the Mullins-Sekerka model with kinetic drag
in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Deng X
(2011)
Global solutions of shock reflection by wedges for the nonlinear wave equation
in Chinese Annals of Mathematics, Series B
Diening L
(2013)
Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology
in SIAM Journal on Numerical Analysis
Ding M
(2012)
Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations
in Zeitschrift für angewandte Mathematik und Physik
Ding M
(2013)
Global existence and non-relativistic global limits of entropy solutions to the 1D piston problem for the isentropic relativistic Euler equations
in Journal of Mathematical Physics
Dolzmann G
(2012)
BMO and uniform estimates for multi-well problems
in Manuscripta Mathematica
Duzaar F
(2007)
The existence of regular boundary points for non-linear elliptic systems
in Journal für die reine und angewandte Mathematik (Crelles Journal)
Fang B
(2016)
The uniqueness of transonic shocks in supersonic flow past a 2-D wedge
in Journal of Mathematical Analysis and Applications
| Description | This was a broad grant designed to help consolidate research on nonlinear partial differential equations in the UK. In particular the Oxford Centre for Nonlinear PDE was founded as a result of the grant and is now a leading international centre. As regards specific research advances, these were in various areas of applications of PDE, for example to fluid and solid mechnaics, liquid crystals, electromagnetism, and relativity. |
| Exploitation Route | Through publications and consultation with current and former members of OxPDE. |
| Sectors | Aerospace Defence and Marine Chemicals Construction Electronics Energy Environment |
| URL | https://www0.maths.ox.ac.uk/groups/oxpde |
| Description | Advanced Investigator Grant |
| Amount | € 2,006,998 (EUR) |
| Funding ID | 291053 |
| Organisation | European Research Council (ERC) |
| Sector | Public |
| Country | Belgium |
| Start | 03/2012 |
| End | 03/2017 |