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

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Helmers M (2012) Kinks in two-phase lipid bilayer membranes in Calculus of Variations and Partial Differential Equations

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Ball J (2010) Local minimizers and planar interfaces in a phase-transition model with interfacial energy in Calculus of Variations and Partial Differential Equations

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Ball J (2017) Partial regularity and smooth topology-preserving approximations of rough domains in Calculus of Variations and Partial Differential Equations

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Chen G (2019) Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing in Chinese Annals of Mathematics, Series B

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Deng X (2011) Global solutions of shock reflection by wedges for the nonlinear wave equation in Chinese Annals of Mathematics, Series B

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Chen G (2015) Weak continuity and compactness for nonlinear partial differential equations in Chinese Annals of Mathematics, Series B

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Chrusciel P (2011) A lower bound for the mass of axisymmetric connected black hole data sets in Classical and Quantum Gravity

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Chen G (2009) Isometric Immersions and Compensated Compactness in Communications in Mathematical Physics

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Chrusciel P (2008) On Mason's Rigidity Theorem in Communications in Mathematical Physics

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Niethammer B (2011) Optimal Bounds for Self-Similar Solutions to Coagulation Equations with Product Kernel in Communications in Partial Differential Equations

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Chen G (2013) Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws in Communications in Partial Differential Equations

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Napoli A (2014) On the validity of the Euler-Lagrange system in Communications on Pure and Applied Analysis

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Torres M (2011) On the structure of solutions of nonlinear hyperbolic systems of conservation laws in Communications on Pure and Applied Analysis

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Kirchheim B (2011) Automatic convexity of rank-1 convex functions in Comptes Rendus Mathematique

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Herrmann M (2011) On selection criteria for problems with moving inhomogeneities in Continuum Mechanics and Thermodynamics

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Feldman M (2016) Transonic flows with shocks past curved wedges for the full euler equations in Discrete and Continuous Dynamical Systems

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Soneji P (2014) Relaxation in BV of integrals with superlinear growth in ESAIM: Control, Optimisation and Calculus of Variations

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Capdeboscq Y (2011) Root growth: homogenization in domains with time dependent partial perforations in ESAIM: Control, Optimisation and Calculus of Variations

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Knezevic D (2008) Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift in ESAIM: Mathematical Modelling and Numerical Analysis

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Barrett J (2010) Finite element approximation of kinetic dilute polymer models with microscopic cut-off in ESAIM: Mathematical Modelling and Numerical Analysis

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Knezevic D (2009) A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model in ESAIM: Mathematical Modelling and Numerical Analysis

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Schwab C (2008) Sparse finite element approximation of high-dimensional transport-dominated diffusion problems in ESAIM: Mathematical Modelling and Numerical Analysis

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Ortner C (2008) Analysis of a quasicontinuum method in one dimension in ESAIM: Mathematical Modelling and Numerical Analysis

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Capdeboscq Y (2008) Imagerie électromagnétique de petites inhomogénéités in ESAIM: Proceedings

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AMMARI H (2009) Mathematical models and reconstruction methods in magneto-acoustic imaging in European Journal of Applied Mathematics

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Figueroa L (2012) Greedy Approximation of High-Dimensional Ornstein-Uhlenbeck Operators in Foundations of Computational Mathematics

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Chrusciel P (2011) Ghost points in inverse scattering constructions of stationary Einstein metrics in General Relativity and Gravitation

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Buffa A (2008) Compact embeddings of broken Sobolev spaces and applications in IMA Journal of Numerical Analysis

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Berrone S (2007) Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows in IMA Journal of Numerical Analysis

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Barrett J (2008) Numerical approximation of corotational dumbbell models for dilute polymers in IMA Journal of Numerical Analysis

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Chen G (2020) Traces and extensions of bounded divergence-measure fields on rough open sets in Indiana University Mathematics Journal

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Kurzke M (2009) Dynamics for Ginzburg-Landau vortices under a mixed flow in Indiana University Mathematics Journal

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Kreisbeck C (2015) Thin-film limits of functionals on A-free vector fields in Indiana University Mathematics Journal

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Ortner C (2011) Stress-based atomistic/continuum coupling: a new variant of the quasicontinuum approximation in International Journal for Multiscale Computational Engineering

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Ammari H (2010) Progress on the strong Eshelby's conjecture and extremal structures for the elastic moment tensor in Journal de Mathématiques Pures et Appliquées

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Bourgain J (2015) On the Morse-Sard property and level sets of W n ,1 Sobolev functions on R n in Journal für die reine und angewandte Mathematik (Crelles Journal)

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Duzaar F (2007) The existence of regular boundary points for non-linear elliptic systems in Journal für die reine und angewandte Mathematik (Crelles Journal)

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Herrmann M (2009) Self-similar solutions for the LSW model with encounters in Journal of Differential Equations

 
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