# 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

Chen G
(2009)

*Evolution of Discontinuity and Formation of Triple-Shock Pattern in Solutions to a Two-Dimensional Hyperbolic System of Conservation Laws*in SIAM Journal on Mathematical Analysis
LANGWALLNER B
(2011)

*EXISTENCE AND CONVERGENCE RESULTS FOR THE GALERKIN APPROXIMATION OF AN ELECTRONIC DENSITY FUNCTIONAL*in Mathematical Models and Methods in Applied Sciences
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
BARRETT J
(2012)

*EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS II: HOOKEAN-TYPE MODELS*in Mathematical Models and Methods in Applied Sciences
Hudson T
(2014)

*Existence and Stability of a Screw Dislocation under Anti-Plane Deformation*in Archive for Rational Mechanics and Analysis
Fuchs M
(2008)

*Existence of global solutions for a parabolic system related to the nonlinear Stokes problem*in Journal of Mathematical Sciences
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
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 Equations
Bulícek M
(2013)

*Existence of Global Weak Solutions to Implicitly Constituted Kinetic Models of Incompressible Homogeneous Dilute Polymers*in Communications in Partial Differential Equations
LARSEN C
(2012)

*EXISTENCE OF SOLUTIONS TO A REGULARIZED MODEL OF DYNAMIC FRACTURE*in Mathematical Models and Methods in Applied Sciences
Makridakis C
(2012)

*Finite Element Analysis of Cauchy-Born Approximations to Atomistic Models*in Archive for Rational Mechanics and Analysis
Barrett J
(2010)

*Finite element approximation of kinetic dilute polymer models with microscopic cut-off*in ESAIM: Mathematical Modelling and Numerical Analysis
Diening L
(2013)

*Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology*in SIAM Journal on Numerical Analysis
Acharya A
(2017)

*Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions*in Archive for Rational Mechanics and Analysis
Chen G
(2018)

*Fluids, geometry, and the onset of Navier-Stokes turbulence in three space dimensions*in Physica D: Nonlinear Phenomena
Chen G
(2021)

*Formation of Singularities and Existence of Global Continuous Solutions for the Compressible Euler Equations*in SIAM Journal on Mathematical Analysis
Chen GQ
(2015)

*Free boundary problems in shock reflection/diffraction and related transonic flow problems.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Bedford S
(2015)

*Function Spaces for Liquid Crystals*in Archive for Rational Mechanics and Analysis
Ball J
(2015)

*Geometry of polycrystals and microstructure*in MATEC Web of Conferences
Chrusciel P
(2011)

*Ghost points in inverse scattering constructions of stationary Einstein metrics*in General Relativity and Gravitation
Ding M
(2013)

*Global existence and non-relativistic global limits of entropy solutions to the 1D piston problem for the isentropic relativistic Euler equations*in Journal of Mathematical Physics
Paicu M
(2011)

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

*Global solutions of shock reflection by wedges for the nonlinear wave equation*in Chinese Annals of Mathematics, Series B
Chen G
(2012)

*Global Steady Subsonic Flows through Infinitely Long Nozzles for the Full Euler Equations*in SIAM Journal on Mathematical Analysis
Chen GG
(2018)

*Global Weak Rigidity of the Gauss-Codazzi-Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity.*in Journal of geometric analysis
Figueroa L
(2012)

*Greedy Approximation of High-Dimensional Ornstein-Uhlenbeck Operators*in Foundations of Computational Mathematics
Carozza M
(2011)

*Higher differentiability of minimizers of convex variational integrals*in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Chapman S
(2016)

*Homogenization of a Row of Dislocation Dipoles from Discrete Dislocation Dynamics*in SIAM Journal on Applied Mathematics
Capdeboscq Y
(2008)

*Imagerie électromagnétique de petites inhomogénéités*in ESAIM: Proceedings
Capdeboscq Y
(2009)

*Imaging by Modification: Numerical Reconstruction of Local Conductivities from Corresponding Power Density Measurements*in SIAM Journal on Imaging Sciences
Capdeboscq Y
(2007)

*Improved Hashin-Shtrikman Bounds for Elastic Moment Tensors and an Application*in Applied Mathematics and Optimization
Ball J
(2015)

*Incompatible Sets of Gradients and Metastability*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
Briane M
(2012)

*Interior Regularity Estimates in High Conductivity Homogenization and Application*in Archive for Rational Mechanics and Analysis
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
Chen G
(2009)

*Isometric Immersions and Compensated Compactness*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
(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
(2019)

*Kolmogorov-type theory of compressible turbulence and inviscid limit of the Navier-Stokes equations in R 3*in Physica D: Nonlinear Phenomena
Herrmann M
(2012)

*Kramers and Non-Kramers Phase Transitions in Many-Particle Systems with Dynamical Constraint*in Multiscale Modeling & Simulation
Giannoulis J
(2008)

*Lagrangian and Hamiltonian two-scale reduction*in Journal of Mathematical Physics
Majumdar A
(2009)

*Landau-De Gennes Theory of Nematic Liquid Crystals: the Oseen-Frank Limit and Beyond*in Archive for Rational Mechanics and Analysis
Chen G
(2009)

*Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations*in Proceedings of the American Mathematical Society
Koch G
(2009)

*Liouville theorems for the Navier-Stokes equations and applications*in Acta Mathematica
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
Ball J
(2010)

*Local minimizers and planar interfaces in a phase-transition model with interfacial energy*in Calculus of Variations and Partial Differential Equations
Owhadi H
(2011)

*Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast*in Multiscale Modeling & Simulation
AMMARI H
(2009)

*Mathematical models and reconstruction methods in magneto-acoustic imaging*in European Journal of Applied Mathematics
Ball J
(2017)

*Mathematics and liquid crystals*in Molecular Crystals and Liquid CrystalsDescription | 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 |