James C. Robinson
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.
If you would like copies of any of the following, please contact me by email.
Theory of inertial manifolds
Manifolds and the Cone Condition, Dynamic Systems and
Applications 2 (1993) 311-330
Manifolds for the Kuramoto-Sivashinsky equation, Physics Letters
A 184 (1994) 190-193
Behavior in Partial Differential Equations (review article),
Chaos 5 (1995) 330-345
Manifolds and the Strong Squeezing Property, Nonlinear Evolution
Equations & Dynamical Systems, NEEDS 94, World Scientific, Singapore
- 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)
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
Accurate Approximate Inertial Manifolds of Fixed Dimension, Physics
Letters A 230 (1997) 301-304
manifolds with and without delay, Discrete & Continuous Dynamical
Systems 5 (1999) 813-824
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
Attractors: Topology and Finite-Dimensional Dynamics, Journal of
Dynamics and Differential Equations 11 (1999) 557-581
- with P.K.
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)
- 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)
- 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.
- 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
- 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.
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.
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.
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.
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.
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