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

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

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

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

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

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

Nguyen L (2012) Refined approximation for minimizers of a Landau-de Gennes energy functional in Calculus of Variations and Partial Differential Equations

Niethammer B (2010) A rigorous derivation of mean-field models for diblock copolymer melts in Calculus of Variations and Partial Differential Equations

Kristensen J (2009) Relaxation of signed integral functionals in BV in Calculus of Variations and Partial Differential Equations

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

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

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

Jones GW (2010) Modelling apical constriction in epithelia using elastic shell theory. in Biomechanics and modeling in mechanobiology

Chen G (2018) Isometric embedding via strongly symmetric positive systems in Asian Journal of Mathematics

Kristensen J (2010) Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV in Archive for Rational Mechanics and Analysis

Chen G (2019) Cauchy Fluxes and Gauss-Green Formulas for Divergence-Measure Fields Over General Open Sets in Archive for Rational Mechanics and Analysis

Bedford S (2015) Function Spaces for Liquid Crystals in Archive for Rational Mechanics and Analysis

Makridakis C (2012) Finite Element Analysis of Cauchy-Born Approximations to Atomistic Models in Archive for Rational Mechanics and Analysis

Paicu M (2011) Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System in Archive for Rational Mechanics and 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