# Analysis of Nonlinear Partial Differential Equations

Lead Research Organisation:
University of Oxford

### 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

Capdeboscq Y
(2009)

*Imaging by Modification: Numerical Reconstruction of Local Conductivities from Corresponding Power Density Measurements*in SIAM Journal on Imaging Sciences
Kristensen J
(2009)

*Relaxation of signed integral functionals in BV*in Calculus of Variations and Partial Differential Equations
Kristensen J
(2010)

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

*Dynamics and self-similarity in min-driven clustering*in Transactions of the American Mathematical Society
Chen G
(2010)

*Vanishing viscosity limit of the Navier-Stokes equations to the euler equations for compressible fluid flow*in Communications on Pure and Applied Mathematics
Ball J
(2010)

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

*A hyperbolic system of conservation laws for fluid flows through compliant axisymmetric vessels*in Acta Mathematica Scientia
Herrmann M
(2010)

*Action Minimising Fronts in General FPU-type Chains*in Journal of Nonlinear Science
Dai S
(2010)

*Crossover in coarsening rates for the monopole approximation of the Mullins-Sekerka model with kinetic drag*in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Burke S
(2010)

*An Adaptive Finite Element Approximation of a Variational Model of Brittle Fracture*in SIAM Journal on Numerical Analysis
Barrett J
(2010)

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

*Boundary Regularity in Variational Problems*in Archive for Rational Mechanics and Analysis
Ball J
(2010)

*Nematic Liquid Crystals: From Maier-Saupe to a Continuum Theory*in Molecular Crystals and Liquid Crystals
Chrusciel P
(2010)

*A Uniqueness Theorem for Degenerate Kerr-Newman Black Holes*in Annales Henri Poincaré
Chrusciel P
(2010)

*On smoothness of black saturns*in Journal of High Energy Physics
Melcher C
(2010)

*Thin-Film Limits for Landau-Lifshitz-Gilbert Equations*in SIAM Journal on Mathematical Analysis
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
Niethammer B
(2010)

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

*Enhanced Resolution in Structured Media*in SIAM Journal on Applied Mathematics
Condette N
(2010)

*Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth*in Mathematics of Computation
Peschka D
(2010)

*Self-similar rupture of viscous thin films in the strong-slip regime*in Nonlinearity
Ball J
(2011)

*Orientability and Energy Minimization in Liquid Crystal Models*in Archive for Rational Mechanics and Analysis
Ortner C
(2011)

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

*Global Existence and Regularity for the Full Coupled Navier-Stokes and Q -Tensor System*in SIAM Journal on Mathematical Analysis
HELMERS M
(2011)

*SNAPPING ELASTIC CURVES AS A ONE-DIMENSIONAL ANALOGUE OF TWO-COMPONENT LIPID BILAYERS*in Mathematical Models and Methods in Applied Sciences
Breit D
(2011)

*Quasiconvex variational functionals in Orlicz-Sobolev spaces*in Annali di Matematica Pura ed Applicata
LANGWALLNER B
(2011)

*EXISTENCE AND CONVERGENCE RESULTS FOR THE GALERKIN APPROXIMATION OF AN ELECTRONIC DENSITY FUNCTIONAL*in Mathematical Models and Methods in Applied Sciences
Niethammer B
(2011)

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

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

*A priori error analysis of two force-based atomistic/continuum models of a periodic chain*in Numerische Mathematik
Owhadi H
(2011)

*Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast*in Multiscale Modeling & Simulation
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
Ammari H
(2011)

*Microwave Imaging by Elastic Deformation*in SIAM Journal on Applied Mathematics
Torres M
(2011)

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

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

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

*Automatic convexity of rank-1 convex functions*in Comptes Rendus Mathematique
Paicu M
(2011)

*Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System*in Archive for Rational Mechanics and Analysis
Niethammer B
(2011)

*Self-similar solutions with fat tails for a coagulation equation with diagonal kernel*in Comptes Rendus Mathematique
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
Choquet-Bruhat Y
(2011)

*The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions*in Annales Henri Poincaré
Carozza M
(2011)

*Higher differentiability of minimizers of convex variational integrals*in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Herrmann M
(2011)

*On selection criteria for problems with moving inhomogeneities*in Continuum Mechanics and Thermodynamics
Chrusciel P
(2011)

*Ghost points in inverse scattering constructions of stationary Einstein metrics*in General Relativity and Gravitation
Chen G
(2011)

*Nonlinear Conservation Laws and Applications*
NEGRI M
(2011)

*QUASI-STATIC CRACK PROPAGATION BY GRIFFITH'S CRITERION*in Mathematical Models and Methods in Applied Sciences
Capdeboscq Y
(2012)

*Numerical computation of approximate generalized polarization tensors*in Applicable Analysis
Barrett J
(2012)

*Existence of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers with variable density and viscosity*in Journal of Differential EquationsDescription | 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 |