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
LARSEN C
(2012)
EXISTENCE OF SOLUTIONS TO A REGULARIZED MODEL OF DYNAMIC FRACTURE
in Mathematical Models and Methods in Applied Sciences
Chen G
(2012)
Global Steady Subsonic Flows through Infinitely Long Nozzles for the Full Euler Equations
in SIAM Journal on Mathematical Analysis
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
Nguyen L
(2012)
Refined approximation for minimizers of a Landau-de Gennes energy functional
in Calculus of Variations and Partial Differential Equations
Niethammer B
(2012)
Self-similar Solutions with Fat Tails for Smoluchowski's Coagulation Equation with Locally Bounded Kernels
in Communications in Mathematical Physics
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
Makridakis C
(2012)
Finite Element Analysis of Cauchy-Born Approximations to Atomistic Models
in Archive for Rational Mechanics and Analysis
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
(2012)
Shallow water equations: viscous solutions and inviscid limit
in Zeitschrift für angewandte Mathematik und Physik
HERRMANN M
(2012)
OSCILLATORY WAVES IN DISCRETE SCALAR CONSERVATION LAWS
in Mathematical Models and Methods in Applied Sciences
Capdeboscq Y
(2012)
Numerical computation of approximate generalized polarization tensors
in Applicable Analysis
Breit D
(2012)
Solenoidal Lipschitz truncation and applications in fluid mechanics
in Journal of 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
Dolzmann G
(2012)
BMO and uniform estimates for multi-well problems
in Manuscripta Mathematica
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
Capdeboscq Y
(2012)
On the scattered field generated by a ball inhomogeneity of constant index
in Asymptotic Analysis
Helmers M
(2012)
Kinks in two-phase lipid bilayer membranes
in Calculus of Variations and Partial Differential Equations
Chen G
(2012)
On Nonlinear Stochastic Balance Laws
in Archive for Rational Mechanics and Analysis
Varvaruca E
(2012)
Equivalence of weak formulations of the steady water waves equations.
in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Herrmann M
(2012)
Kramers and Non-Kramers Phase Transitions in Many-Particle Systems with Dynamical Constraint
in Multiscale Modeling & Simulation
Briane M
(2012)
Interior Regularity Estimates in High Conductivity Homogenization and Application
in Archive for Rational Mechanics and Analysis
Gallagher I
(2012)
A profile decomposition approach to the $$L^\infty _t(L^{3}_x)$$ Navier-Stokes regularity criterion
in Mathematische Annalen
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
BURKE S
(2013)
AN ADAPTIVE FINITE ELEMENT APPROXIMATION OF A GENERALIZED AMBROSIO-TORTORELLI FUNCTIONAL
in Mathematical Models and Methods in Applied Sciences
Ball JM
(2013)
Entropy and convexity for nonlinear partial differential equations.
in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Chen G
(2013)
Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System
in Archive for Rational Mechanics and Analysis
Capdeboscq Y
(2013)
On one-dimensional inverse problems arising from polarimetric measurements of nematic liquid crystals
in Inverse Problems
Chen G
(2013)
Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws
in Communications in Partial Differential Equations
Diening L
(2013)
Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology
in SIAM Journal on Numerical Analysis
Bulícek M
(2013)
Existence of Global Weak Solutions to Implicitly Constituted Kinetic Models of Incompressible Homogeneous Dilute Polymers
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 Physik
Bourgain J
(2013)
On the Morse-Sard property and level sets of Sobolev and BV functions
in Revista Matemática Iberoamericana
Chen G
(2013)
Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows
in Journal of Mathematical Physics
Napoli A
(2014)
On the validity of the Euler-Lagrange system
in Communications on Pure and Applied Analysis
Helmers M
(2014)
CONVERGENCE OF AN APPROXIMATION FOR ROTATIONALLY SYMMETRIC TWO-PHASE LIPID BILAYER MEMBRANES
in The Quarterly Journal of Mathematics
Ball J
(2014)
Quasistatic Nonlinear Viscoelasticity and Gradient Flows
in Journal of Dynamics and Differential Equations
Soneji P
(2014)
Relaxation in BV of integrals with superlinear growth
in ESAIM: Control, Optimisation and Calculus of Variations
Rindler F
(2014)
A local proof for the characterization of Young measures generated by sequences in BV
in Journal of Functional Analysis
Ball J
(2014)
An investigation of non-planar austenite-martensite interfaces
in Mathematical Models and Methods in Applied Sciences
Braides A
(2014)
An Example of Non-Existence of Plane-Like Minimizers for an Almost-Periodic Ising System
in Journal of Statistical Physics
Mielke A
(2014)
An Approach to Nonlinear Viscoelasticity via Metric Gradient Flows
in SIAM Journal on Mathematical Analysis
Choi K
(2014)
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations
in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Bulícek M
(2014)
Analysis and approximation of a strain-limiting nonlinear elastic model
in Mathematics and Mechanics of Solids
Hudson T
(2014)
Existence and Stability of a Screw Dislocation under Anti-Plane Deformation
in Archive for Rational Mechanics and Analysis
Hudson T
(2015)
Analysis of Stable Screw Dislocation Configurations in an Antiplane Lattice Model
in SIAM Journal on Mathematical Analysis
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
(2015)
Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations
in Archive for Rational Mechanics and Analysis
Chen G
(2015)
Weak continuity and compactness for nonlinear partial differential equations
in Chinese Annals of Mathematics, Series B
Ball J
(2015)
Incompatible Sets of Gradients and Metastability
in Archive for Rational Mechanics and Analysis
Description | 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 | 04/2012 |
End | 03/2017 |