# 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

Schwab C
(2008)

*Sparse finite element approximation of high-dimensional transport-dominated diffusion problems*in ESAIM: Mathematical Modelling and Numerical Analysis
Condette N
(2010)

*Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth*in Mathematics of Computation
Knezevic D
(2008)

*Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift*in ESAIM: Mathematical Modelling and Numerical Analysis
Chen G
(2020)

*Stability of Multidimensional Thermoelastic Contact Discontinuities*in Archive for Rational Mechanics and Analysis
Chen G
(2013)

*Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows*in Journal of Mathematical Physics
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
Chen G
(2017)

*Stability of transonic shocks in steady supersonic flow past multidimensional wedges*in Advances in Mathematics
Chen G
(2019)

*Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles*in Advances in Mathematics
Ortner C
(2011)

*Stress-based atomistic/continuum coupling: a new variant of the quasicontinuum approximation*in International Journal for Multiscale Computational Engineering
Chen G
(2015)

*Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations*in Archive for Rational Mechanics and Analysis
Choquet-Bruhat Y
(2011)

*The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions*in Annales Henri Poincaré
Duzaar F
(2007)

*The existence of regular boundary points for non-linear elliptic systems*in Journal für die reine und angewandte Mathematik (Crelles Journal)
Chen G
(2009)

*The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries*in Acta Mathematica Scientia
Beretta E
(2009)

*Thin cylindrical conductivity inclusions in a three-dimensional domain: a polarization tensor and unique determination from boundary data*in Inverse Problems
Melcher C
(2010)

*Thin-Film Limits for Landau-Lifshitz-Gilbert Equations*in SIAM Journal on Mathematical Analysis
Peschka D
(2009)

*Thin-film rupture for large slip*in Journal of Engineering Mathematics
Chrusciel P
(2009)

*Topological Censorship for Kaluza-Klein Space-Times*in Annales Henri Poincaré
Allaire G
(2009)

*Two asymptotic models for arrays of underground waste containers*in Applicable Analysis
Berrone S
(2007)

*Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows*in IMA Journal of Numerical Analysis
Chen G
(2009)

*Uniqueness of transonic shock solutions in a duct for steady potential flow*in Journal of Differential Equations
Chen G
(2018)

*Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows*in Archive for Rational Mechanics and 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
Chen G
(2015)

*Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data*in Communications in Mathematical Physics
Capella A
(2007)

*Wave-type dynamics in ferromagnetic thin films and the motion of Néel walls*in Nonlinearity
Chen G
(2015)

*Weak continuity and compactness for nonlinear partial differential equations*in Chinese Annals of Mathematics, Series B
Chen G
(2021)

*Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity*in Archive for Rational Mechanics and Analysis
Chen G
(2009)

*Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding*in Proceedings of the American Mathematical Society
Chen G
(2013)

*Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws*in Communications in Partial Differential Equations
Chen G
(2013)

*Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows*in Zeitschrift für angewandte Mathematik und PhysikDescription | 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 |