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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chen G (2009) Uniqueness of transonic shock solutions in a duct for steady potential flow in Journal of Differential Equations

Breit D (2012) Solenoidal Lipschitz truncation and applications in fluid mechanics in Journal of Differential Equations

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

Ball J (2014) Quasistatic Nonlinear Viscoelasticity and Gradient Flows in Journal of Dynamics and Differential Equations

Peschka D (2009) Thin-film rupture for large slip in Journal of Engineering Mathematics

Chrusciel P (2010) On smoothness of black saturns in Journal of High Energy Physics

Giannoulis J (2008) Lagrangian and Hamiltonian two-scale reduction in Journal of Mathematical Physics

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