# 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 OptimizationDescription | 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 |