# James C. Robinson

email: j.c.robinson@warwick.ac.uk

MASDOC A1: LINEAR PARTIAL DIFFERENTIAL EQUATIONS

MASDOC: The current version of the A1 notes (9/12/10) are here. Please let me know of any errors/incosistencies. The old Functional Analysis I webpage may be useful.

Examples sheets: One, Two, and Three, Four, Five.

Assessed questions should be with be by the end of Wednesday 12th January. [Sheet 1: 3,6, and 12; Sheet 2: 1, 8, and 13; Sheet 3: 4, 7, and 10; Sheets 4 & 5 as starred on the sheet.]

ASSESSED QUESTIONS: CORRECTIONS & HINTS

Sheet 1: Q12 should read $(Au,v)=(u,A^*v)$

Sheet 2: Q13 do not forget to check that you really have the weak derivative

Sheet 3: Q7 For correct formulation of this question see the solutions

Sheet 4: Q3 Should be $|\alpha|=k$ rather than $|\alpha|=k+1$; Q4 find the weak form by taking the inner product with some $\phi\in C^\infty(\Omega)$; Q5 $\lambda_0=0$

Sheet 5: Do Q3 rather than Q1

SOLUTIONS to the problems (apart from the assessed questions!) are here.

ORALS will be on Monday 10th January (first day of term). Please sign up on the sheet just outside my office (C2.20). Here are some guidelines to tell you what to expect.

The Infinite-dimensional dynamical systems (2007 course) lecture notes are here.

PUBLICATIONS

Theory of inertial manifolds

• Inertial Manifolds and the Cone Condition, Dynamic Systems and Applications 2 (1993) 311-330
• Inertial Manifolds for the Kuramoto-Sivashinsky equation, Physics Letters A 184 (1994) 190-193
• Finite-Dimensional Behavior in Partial Differential Equations (review article), Chaos 5 (1995) 330-345
• Inertial Manifolds and the Strong Squeezing Property, Nonlinear Evolution Equations & Dynamical Systems, NEEDS 94, World Scientific, Singapore (1995) 178-187
• A concise proof of the "geometric" construction of inertial manifolds, Physics Letters A 200 (1995) 415-417
• The Asymptotic Completeness of Inertial Manifolds, Nonlinearity 9 (1996) 1325-1340
• Convergent Families of Inertial Manifolds for Convergent Approximations, Numerical Algorithms 14 (1997) 179-188
• Some Closure Results for Inertial Manifolds, Journal of Dynamics and Differential Equations 9 (1997) 373-400
• Arbitrarily Accurate Approximate Inertial Manifolds of Fixed Dimension, Physics Letters A 230 (1997) 301-304
• Inertial manifolds with and without delay, Discrete & Continuous Dynamical Systems 5 (1999) 813-824
• Computing inertial manifolds, Discrete & Continuous Dynamical Systems 8 (2002) 815--833.

Deterministic theory of global attractors

• with J.A. Langa: Determining Asymptotic Behaviour from the Dynamics on Attracting Sets, Journal of Dynamics and Differential Equations 11 (1999) 319-331
• Global Attractors: Topology and Finite-Dimensional Dynamics, Journal of Dynamics and Differential Equations 11 (1999) 557-581
• with P.K. Friz: Smooth attractors have zero "thickness", Mathematical Analaysis and Applications 240 (1999) 37-46
• with P.K. Friz: Parametrising the attractor of the two-dimensional Navier-Stokes equations with a finite set of nodal values , Physica D148 (2001) 201-220
• with P.K. Friz & I. Kukavica: Nodal parametrisation of analytic attractors, Discrete and Continuous Dynamical Systems 7 (2001) 643-657.
• A rigorous treatment of 'experimental' observations for the two-dimensional Navier-Stokes equations, Royal Society of London Proceedings A 457 (2001) 1007-1020.
• with J.A. Langa: A finite number of points observations which determine a non-autonomous fluid flow, Nonlinearity 14 (2001) 673-682.
• Low-dimensional attractors arise from forcing at small scales, Physica D 181 (2003) 39-44.
• with I. Kukavica: Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem, Physica D 196 (2004) 45-66
• with O.M. Tearne: Boundaries of attractors as omega limit sets, Stochastics & Dynamics 5 (2005) 97-109
• A topological delay embedding theorem for infinite-dimensional dynamical systems, Nonlinearity 18 (2005) 2135-2143

Deterministic infinite-dimensional dynamical systems

• with A. Vidal-Lopez: Minimal periods of semilinear evolution equations with Lipschitz nonlinearity, Journal of Differential Equations 220 (2006) 396-406

Rigorous analysis of physical models

• A coupled particle-continuum model: well-posedness and the limit of zero radius, Royal Society of London Proceedings A 460 (2004) 1311-1334

Random attractors and random dynamical systems

• with T. Caraballo & J.A. Langa: Upper Semicontinuity of Attractors for Small Random Perturbations of Dynamical Systems, Communications in PDEs 23 (1998) 1557-1581
• with T. Caraballo & J.A. Langa: Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete and Continuous Dynamical Systems 6 (2000) 875-892
• with J.A. Langa & T. Caraballo: A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proceedings of the Royal Society of London Series A 457 (2001) 2041-2061.
• Stability of random attractors under perturbation and approximation, Journal of Differential Equations 186 (2002) 652-669
• with P. Marin-Rubio: Attractors for the stochastic 3d Navier-Stokes equations, Stochastics and Dynamics 3 (2003) 279-297
• with T. Caraballo: Stabilisation of linear PDEs by Stratonovich noise, Systems and Control Letters 53 (2004) 41-50
• with J.A.Langa: Fractal dimension of a random invariant set, Journal de Mathematiques Pures et Appliques, to appear.
• with T. Caraballo, J.A.Langa, & H. Crauel: The effecto fo noise on the Chafee-Infante equation: a nonlinear case study, Proceedings of the American Mathematical Society, to appear.

Non-autonomous systems

• with T. Caraballo and J.A.Langa: Attractors for Differential Equations with Variable Delays, Journal of Mathematical Analysis and Applications 260 (2001) 421-438
• with J.A. Langa & A. Suarez: Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity 15 (2002) 887-903
• with J.A. Langa & A. Suarez: Pullback permanence in a non-autonomous competitive Lotka-Volterra model, Journal of Differential Equations 190 (2003) 241--238.
• with J.A. Langa & A. Suarez: Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system, Nonlinearity 16 (2003) 1277-1293.
• with J.A. Langa & A. Suárez: Bifurcation from zero of a complete trajectory for non-autonomous logistic PDEs, International Journal of Bifurcation & Chaos 8 (2005) 2663-2669
• with J.A. Langa & A. Suárez: Bifurcations in non-autonomous scalar ODEs, Journal of Differential Equations, to appear.

Various topics in ODEs

• All possible chaotic dynamics can be approximated in three dimensions, Nonlinearity 11 (1998) 529-545
• Solutions of continuous ODEs obtained as the limit of solutions of Lipschitz ODEs, Nonlinearity 12 (1999) 555-561

Miscellaneous

• with P.K. Friz: Constructing an elementary measure on a space of projections, Journal of Mathematical Analysis and Applications 267 (2002) 714-725
• with R.B. Hoyle: League tables and school effectiveness: a mathematical model, Royal Society of London Proceedings B 270 (2003) 113-119
• with P. Marin Rubio:A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Analysis 11 (2003) 297-322.

BOOKS

• with P.A. Glendinning (eds.): From Finite to Infinite Dimensional Dynamical Systems, proceedings of the NATO ASI workshop at the Isaac Newton Institute, August 1995. Kluwer Academic, 2001.

• Infinite-dimensional dynamical systems, Cambridge University Press "Texts in Applied Mathematics" Series (2001).

• An Introduction to Ordinary Differential Equations, Cambridge University Press 2004.

• with J.L. Rodrigo (eds.): Partial Differential Equations and Fluid Mechanics, LMS Lecture Note Series 264, Cambridge University Press 2009.

• Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics 186, Cambridge University Press 2011.

• LECTURE NOTES

• Some approaches to finite-dimensional behaviour in the Navier-Stokes Equations, lectures given at the University of Seville, Easter 1997.
• Topics in Mathematical Phsyics: a 4th year lecture course at Warwick based on my book, Infinite-Dimensional Dynamical Systems.
• Attractors and finite-dimensional behaviour in the Navier-Stokes equations, notes for lectures given at the ICMS Instructional Conference on Mathematical Hydrodynamics, June 2003.
• The Navier-Stokes equations, Lectures given at the AARMS Summer school at Memorial University, Newfoundland, July/August 2003