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
Braides A
(2016)
Quasi-static damage evolution and homogenization: A case study of non-commutability
in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
NEGRI M
(2011)
QUASI-STATIC CRACK PROPAGATION BY GRIFFITH'S CRITERION
in Mathematical Models and Methods in Applied Sciences
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
Ball J
(2017)
Partial regularity and smooth topology-preserving approximations of rough domains
in Calculus of Variations and Partial Differential Equations
HERRMANN M
(2012)
OSCILLATORY WAVES IN DISCRETE SCALAR CONSERVATION LAWS
in Mathematical Models and Methods in Applied Sciences
Ball J
(2008)
Orientable and Non-Orientable Line Field Models for Uniaxial Nematic Liquid Crystals
in Molecular Crystals and Liquid Crystals
Ball J
(2011)
Orientability and Energy Minimization in Liquid Crystal Models
in Archive for Rational Mechanics and Analysis
Rindler F
(2009)
Optimal Control for Nonconvex Rate-Independent Evolution Processes
in SIAM Journal on Control and Optimization
Niethammer B
(2011)
Optimal Bounds for Self-Similar Solutions to Coagulation Equations with Product Kernel
in Communications in Partial Differential Equations
BALL J
(2012)
ON UNIQUENESS FOR TIME HARMONIC ANISOTROPIC MAXWELL'S EQUATIONS WITH PIECEWISE REGULAR COEFFICIENTS
in Mathematical Models and Methods in Applied Sciences
Napoli A
(2014)
On the validity of the Euler-Lagrange system
in Communications on Pure and Applied Analysis
Torres M
(2011)
On the structure of solutions of nonlinear hyperbolic systems of conservation laws
in Communications on Pure and Applied Analysis
Capdeboscq Y
(2012)
On the scattered field generated by a ball inhomogeneity of constant index
in Asymptotic Analysis
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)
Bourgain J
(2013)
On the Morse-Sard property and level sets of Sobolev and BV functions
in Revista Matemática Iberoamericana
Chrusciel P
(2010)
On smoothness of black saturns
in Journal of High Energy Physics
Herrmann M
(2011)
On selection criteria for problems with moving inhomogeneities
in Continuum Mechanics and Thermodynamics
Niethammer B
(2007)
On Screening Induced Fluctuations in Ostwald Ripening
in Journal of Statistical Physics
Capdeboscq Y
(2013)
On one-dimensional inverse problems arising from polarimetric measurements of nematic liquid crystals
in Inverse Problems
Chen G
(2012)
On Nonlinear Stochastic Balance Laws
in Archive for Rational Mechanics and Analysis
Chrusciel P
(2008)
On Mason's Rigidity Theorem
in Communications in Mathematical Physics
Chrusciel P
(2009)
On higher dimensional black holes with Abelian isometry group
in Journal of Mathematical Physics
Capdeboscq Y
(2012)
Numerical computation of approximate generalized polarization tensors
in Applicable Analysis
Barrett J
(2008)
Numerical approximation of corotational dumbbell models for dilute polymers
in IMA Journal of Numerical Analysis
Chen G
(2018)
Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
in Archive for Rational Mechanics and Analysis
Chen G
(2011)
Nonlinear Conservation Laws and Applications
Ball J
(2010)
Nematic Liquid Crystals: From Maier-Saupe to a Continuum Theory
in Molecular Crystals and Liquid Crystals
Jones G
(2009)
Modelling apical constriction in epithelia using elastic shell theory
in Biomechanics and Modeling in Mechanobiology
Ammari H
(2011)
Microwave Imaging by Elastic Deformation
in SIAM Journal on Applied Mathematics
Ball J
(2017)
Mathematics and liquid crystals
in Molecular Crystals and Liquid Crystals
AMMARI H
(2009)
Mathematical models and reconstruction methods in magneto-acoustic imaging
in European Journal of Applied Mathematics
Owhadi H
(2011)
Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast
in Multiscale Modeling & Simulation
Ball J
(2010)
Local minimizers and planar interfaces in a phase-transition model with interfacial energy
in Calculus of Variations and Partial Differential Equations
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
Koch G
(2009)
Liouville theorems for the Navier-Stokes equations and applications
in Acta Mathematica
Chen G
(2009)
Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations
in Proceedings of the American Mathematical Society
Majumdar A
(2009)
Landau-De Gennes Theory of Nematic Liquid Crystals: the Oseen-Frank Limit and Beyond
in Archive for Rational Mechanics and Analysis
Giannoulis J
(2008)
Lagrangian and Hamiltonian two-scale reduction
in Journal of Mathematical Physics
Herrmann M
(2012)
Kramers and Non-Kramers Phase Transitions in Many-Particle Systems with Dynamical Constraint
in Multiscale Modeling & Simulation
Chen G
(2019)
Kolmogorov-type theory of compressible turbulence and inviscid limit of the Navier-Stokes equations in R 3
in Physica D: Nonlinear Phenomena
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
Helmers M
(2012)
Kinks in two-phase lipid bilayer membranes
in Calculus of Variations and Partial Differential Equations
Chen G
(2009)
Isometric Immersions and Compensated Compactness
in Communications in Mathematical Physics
Serra Cassano F
(2015)
Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation
in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Briane M
(2012)
Interior Regularity Estimates in High Conductivity Homogenization and Application
in Archive for Rational Mechanics and Analysis
Chen G
(2016)
Incompressible limit of solutions of multidimensional steady compressible Euler equations
in Zeitschrift für angewandte Mathematik und Physik
Ball J
(2015)
Incompatible Sets of Gradients and Metastability
in Archive for Rational Mechanics and Analysis
Capdeboscq Y
(2007)
Improved Hashin-Shtrikman Bounds for Elastic Moment Tensors and an Application
in Applied Mathematics and Optimization
| 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 |