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

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.

- 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) 2135-2143Deterministic infinite-dimensional dynamical systems - with A. Vidal-Lopez: Minimal periods of semilinear evolution equations with Lipschitz nonlinearity,
*Journal of Differential Equations***220**(2006) 396-406Rigorous 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) 1311-1334Random 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) 555-561Miscellaneous - 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.

If you would like copies of any of the following, please contact me by email.

BOOKS

LECTURE NOTES