# 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

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
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
Chrusciel P
(2011)

*A lower bound for the mass of axisymmetric connected black hole data sets*in Classical and Quantum Gravity
Niethammer B
(2012)

*Self-similar Solutions with Fat Tails for Smoluchowski's Coagulation Equation with Locally Bounded Kernels*in Communications in Mathematical Physics
Chen G
(2015)

*Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data*in Communications in Mathematical Physics
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
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
Bulícek M
(2013)

*Existence of Global Weak Solutions to Implicitly Constituted Kinetic Models of Incompressible Homogeneous Dilute Polymers*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
Chen G
(2013)

*Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws*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
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
Kirchheim B
(2011)

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

*Self-similar solutions with fat tails for a coagulation equation with diagonal kernel*in Comptes Rendus. Mathématique
Herrmann M
(2009)

*Self-similar solutions with fat tails for a coagulation equation with nonlocal drift*in Comptes Rendus. Mathématique
Herrmann M
(2011)

*On selection criteria for problems with moving inhomogeneities*in Continuum Mechanics and Thermodynamics
Chen G
(2008)

*Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone*in Discrete and Continuous Dynamical Systems
Feldman M
(2016)

*Transonic flows with shocks past curved wedges for the full euler equations*in Discrete and Continuous Dynamical Systems
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
Knezevic D
(2008)

*Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift*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
(2009)

*A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model*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
Ortner C
(2008)

*Analysis of a quasicontinuum method in one dimension*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
Buffa A
(2008)

*Compact embeddings of broken Sobolev spaces and applications*in IMA Journal of Numerical Analysis
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
Chen G
(2020)

*Traces and extensions of bounded divergence-measure fields on rough open sets*in Indiana University Mathematics Journal
Kurzke M
(2009)

*Dynamics for Ginzburg-Landau vortices under a mixed flow*in Indiana University Mathematics Journal
Kreisbeck C
(2015)

*Thin-film limits of functionals on A-free vector fields*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
Bae M
(2008)

*Regularity of solutions to regular shock reflection for potential flow*in Inventiones mathematicae
Beretta E
(2009)

*Thin cylindrical conductivity inclusions in a three-dimensional domain: a polarization tensor and unique determination from boundary data*in Inverse Problems
Capdeboscq Y
(2013)

*On one-dimensional inverse problems arising from polarimetric measurements of nematic liquid crystals*in Inverse Problems
Chen G
(2022)

*On asymptotic rigidity and continuity problems in nonlinear elasticity on manifolds and hypersurfaces*in Journal de Mathématiques Pures et Appliquées
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 W n ,1 Sobolev functions on R n*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)
Herrmann M
(2009)

*Self-similar solutions for the LSW model with encounters*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 |