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
HERRMANN M
(2012)
OSCILLATORY WAVES IN DISCRETE SCALAR CONSERVATION LAWS
in Mathematical Models and Methods in Applied Sciences
LARSEN C
(2012)
EXISTENCE OF SOLUTIONS TO A REGULARIZED MODEL OF DYNAMIC FRACTURE
in Mathematical Models and Methods in Applied Sciences
Dolzmann G
(2012)
BMO and uniform estimates for multi-well problems
in Manuscripta Mathematica
Capdeboscq Y
(2012)
On the scattered field generated by a ball inhomogeneity of constant index
in Asymptotic Analysis
Varvaruca E
(2012)
Equivalence of weak formulations of the steady water waves equations.
in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Chen G
(2012)
On Nonlinear Stochastic Balance Laws
in Archive for Rational Mechanics and Analysis
Niethammer B
(2012)
Self-similar Solutions with Fat Tails for Smoluchowski's Coagulation Equation with Locally Bounded Kernels
in Communications in Mathematical Physics
Capdeboscq Y
(2012)
Numerical computation of approximate generalized polarization tensors
in Applicable Analysis
Iyer G
(2012)
Coercivity and stability results for an extended Navier-Stokes system
in Journal of Mathematical Physics
BALL J
(2012)
ON UNIQUENESS FOR TIME HARMONIC ANISOTROPIC MAXWELL'S EQUATIONS WITH PIECEWISE REGULAR COEFFICIENTS
in Mathematical Models and Methods in Applied Sciences
Gallagher I
(2012)
A profile decomposition approach to the $$L^\infty _t(L^{3}_x)$$ Navier-Stokes regularity criterion
in Mathematische Annalen
Herrmann M
(2012)
Kramers and Non-Kramers Phase Transitions in Many-Particle Systems with Dynamical Constraint
in Multiscale Modeling & Simulation
Capdeboscq Y
(2011)
On the scattered field generated by a ball inhomogeneity of constant index
Makridakis C
(2011)
A priori error analysis of two force-based atomistic/continuum models of a periodic chain
in Numerische Mathematik
Ammari H
(2011)
Microwave Imaging by Elastic Deformation
in SIAM Journal on Applied Mathematics
Figueroa L
(2011)
Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators
Niethammer B
(2011)
Optimal Bounds for Self-Similar Solutions to Coagulation Equations with Product Kernel
in Communications in Partial Differential Equations
Niethammer B
(2011)
Self-similar solutions with fat tails for a coagulation equation with diagonal kernel
in Comptes Rendus Mathematique
Breit D
(2011)
Quasiconvex variational functionals in Orlicz-Sobolev spaces
in Annali di Matematica Pura ed Applicata
BARRETT J
(2011)
EXISTENCE OF GLOBAL WEAK SOLUTIONS TO DUMBBELL MODELS FOR DILUTE POLYMERS WITH MICROSCOPIC CUT-OFF
in Mathematical Models and Methods in Applied Sciences
LANGWALLNER B
(2011)
EXISTENCE AND CONVERGENCE RESULTS FOR THE GALERKIN APPROXIMATION OF AN ELECTRONIC DENSITY FUNCTIONAL
in Mathematical Models and Methods in Applied Sciences
HELMERS M
(2011)
SNAPPING ELASTIC CURVES AS A ONE-DIMENSIONAL ANALOGUE OF TWO-COMPONENT LIPID BILAYERS
in Mathematical Models and Methods in Applied Sciences
NEGRI M
(2011)
QUASI-STATIC CRACK PROPAGATION BY GRIFFITH'S CRITERION
in Mathematical Models and Methods in Applied Sciences
BARRETT J
(2011)
EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS I: FINITELY EXTENSIBLE NONLINEAR BEAD-SPRING CHAINS
in Mathematical Models and Methods in Applied Sciences
Chen G
(2011)
On Nonlinear Stochastic Balance Laws
Kirchheim B
(2011)
Automatic convexity of rank-1 convex functions
in Comptes Rendus. Mathématique
Capdeboscq Y
(2011)
Numerical Computation of approximate Generalized Polarization Tensors
Owhadi H
(2011)
Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast
in Multiscale Modeling & Simulation
McLeod J
(2011)
Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel
in Journal of Statistical Physics
Chen G
(2011)
Nonlinear Conservation Laws and Applications
Paicu M
(2011)
Global Existence and Regularity for the Full Coupled Navier-Stokes and Q -Tensor System
in SIAM Journal on Mathematical Analysis
Ortner C
(2011)
Stress-based atomistic/continuum coupling: a new variant of the quasicontinuum approximation
in International Journal for Multiscale Computational Engineering
Chrusciel P
(2011)
Ghost points in inverse scattering constructions of stationary Einstein metrics
in General Relativity and Gravitation
Paicu M
(2011)
Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System
in Archive for Rational Mechanics and Analysis
Capdeboscq Y
(2011)
Root growth: homogenization in domains with time dependent partial perforations
in ESAIM: Control, Optimisation and Calculus of Variations
Deng X
(2011)
Global solutions of shock reflection by wedges for the nonlinear wave equation
in Chinese Annals of Mathematics, Series B
Chrusciel P
(2011)
A lower bound for the mass of axisymmetric connected black hole data sets
in Classical and Quantum Gravity
Torres M
(2011)
On the structure of solutions of nonlinear hyperbolic systems of conservation laws
in Communications on Pure and Applied Analysis
| 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 |