# 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

Ball J.M.
(2009)

*An analysis of non-classical austenite-martensite interfaces in CuAlNi*in Proceedings of the International Conference on Martensitic Transformations, ICOMAT-08
Chen G
(2009)

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

*Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding*in Proceedings of the American Mathematical Society
Chen G
(2018)

*Fluids, geometry, and the onset of Navier-Stokes turbulence in three space dimensions*in Physica D: Nonlinear Phenomena
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 GQ
(2015)

*Free boundary problems in shock reflection/diffraction and related transonic flow problems.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Varvaruca E
(2012)

*Equivalence of weak formulations of the steady water waves equations.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Ball JM
(2013)

*Entropy and convexity for nonlinear partial differential equations.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Makridakis C
(2011)

*A priori error analysis of two force-based atomistic/continuum models of a periodic chain*in Numerische Mathematik
Peschka D
(2010)

*Self-similar rupture of viscous thin films in the strong-slip regime*in Nonlinearity
Capella A
(2007)

*Wave-type dynamics in ferromagnetic thin films and the motion of Néel walls*in Nonlinearity
Mielke A
(2009)

*Reverse Approximation of Energetic Solutions to Rate-Independent Processes*in Nonlinear Differential Equations and Applications NoDEA
Tsering-Xiao B
(2020)

*Regularity of solutions to time-harmonic Maxwell's system with various lower than Lipschitz coefficients*in Nonlinear Analysis
Owhadi H
(2011)

*Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast*in Multiscale Modeling & Simulation
Herrmann M
(2012)

*Kramers and Non-Kramers Phase Transitions in Many-Particle Systems with Dynamical Constraint*in Multiscale Modeling & Simulation
Ball J
(2015)

*Discontinuous Order Parameters in Liquid Crystal Theories*in Molecular Crystals and Liquid Crystals
Ball J
(2008)

*Orientable and Non-Orientable Line Field Models for Uniaxial Nematic Liquid Crystals*in Molecular Crystals and Liquid Crystals
Ball J
(2010)

*Nematic Liquid Crystals: From Maier-Saupe to a Continuum Theory*in Molecular Crystals and Liquid Crystals
Ball J
(2017)

*Mathematics and liquid crystals*in Molecular Crystals and Liquid Crystals
Kristensen J
(2008)

*Regularity in oscillatory nonlinear elliptic systems*in Mathematische Zeitschrift
Gallagher I
(2012)

*A profile decomposition approach to the $$L^\infty _t(L^{3}_x)$$ Navier-Stokes regularity criterion*in Mathematische Annalen
Condette N
(2010)

*Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth*in Mathematics of Computation
Bulícek M
(2014)

*Analysis and approximation of a strain-limiting nonlinear elastic model*in Mathematics and Mechanics of Solids
Braides A
(2016)

*Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case*in Mathematics and Mechanics of Solids
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
(2011)

*EXISTENCE OF GLOBAL WEAK SOLUTIONS TO DUMBBELL MODELS FOR DILUTE POLYMERS WITH MICROSCOPIC CUT-OFF*in Mathematical Models and Methods in Applied Sciences
BURKE S
(2013)

*AN ADAPTIVE FINITE ELEMENT APPROXIMATION OF A GENERALIZED AMBROSIO-TORTORELLI FUNCTIONAL*in Mathematical Models and Methods in Applied Sciences
NEGRI M
(2011)

*QUASI-STATIC CRACK PROPAGATION BY GRIFFITH'S CRITERION*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
Barrett J
(2016)

*Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers*in Mathematical Models and Methods in Applied Sciences
BALL J
(2012)

*ON UNIQUENESS FOR TIME HARMONIC ANISOTROPIC MAXWELL'S EQUATIONS WITH PIECEWISE REGULAR COEFFICIENTS*in Mathematical Models and Methods in Applied Sciences
Ball J
(2014)

*An investigation of non-planar austenite-martensite interfaces*in Mathematical Models and Methods in Applied Sciences
HELMERS M
(2011)

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

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

*OSCILLATORY WAVES IN DISCRETE SCALAR CONSERVATION LAWS*in Mathematical Models and Methods in Applied Sciences
LARSEN C
(2012)

*EXISTENCE OF SOLUTIONS TO A REGULARIZED MODEL OF DYNAMIC FRACTURE*in Mathematical Models and Methods in Applied Sciences
Ball J
(2015)

*A probabilistic model for martensitic avalanches*in MATEC Web of Conferences
Ball J
(2015)

*Geometry of polycrystals and microstructure*in MATEC Web of Conferences
Dolzmann G
(2012)

*BMO and uniform estimates for multi-well problems*in Manuscripta Mathematica
Braides A
(2014)

*An Example of Non-Existence of Plane-Like Minimizers for an Almost-Periodic Ising System*in Journal of Statistical Physics
McLeod J
(2011)

*Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel*in Journal of Statistical Physics
Niethammer B
(2007)

*On Screening Induced Fluctuations in Ostwald Ripening*in Journal of Statistical Physics
Herrmann M
(2010)

*Action Minimising Fronts in General FPU-type Chains*in Journal of Nonlinear Science
Fuchs M
(2008)

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

*Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows*in Journal of Mathematical Physics
Chrusciel P
(2009)

*On higher dimensional black holes with Abelian isometry group*in Journal of Mathematical Physics
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
Iyer G
(2012)

*Coercivity and stability results for an extended Navier-Stokes system*in Journal of Mathematical Physics
Giannoulis J
(2008)

*Lagrangian and Hamiltonian two-scale reduction*in Journal of Mathematical Physics
Fang B
(2016)

*The uniqueness of transonic shocks in supersonic flow past a 2-D wedge*in Journal of Mathematical Analysis and ApplicationsDescription | 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 |