Topics in Homogenization of Differential Equations

This course is part of the Taught Course Centre. It is taught during the academic year 2016 – 2017.

Lectures: Fridays 11am – 1pm.

Start date: April 28, 2017.

End date: June 16, 2017.

Lecture room: 6M42 in Imperial college (live broadcast via TCC network).

Total number of hours: 16 (8 sessions).

COURSE MATERIAL

Overview file of the mini-course.

(1) Session No.1 (April 28, 2017):

(2) Session No.2 (May 5, 2017):

(3) Session No.3 (May 12, 2017):

(4) Session No.4 (May 19, 2017):

(5) Session No.5 (May 26, 2017):

(6) Session No.6 (June 2, 2017):

(7) Session No.7 (June 9, 2017):

(8) Session No.8 (June 16, 2017):
Tentative plan for this session: Nonlocal effects induced by homogenization.

This course deals with the modern mathematical analysis of differential equations with highly heterogeneous coefficients — modelling multiple scale physical phenomena in continuum mechanics. Direct numerical computation of solutions to such models can be very costly or even out of reach in many scenarios. Mathematical theory of homogenization suggests the “averaging” of heterogeneities in the multiscale models to deduce some macroscopic behaviour. In the last three decades, the mathematical study of homogenization and related topics has seen tremendous progress with far-reaching consequences in engineering sciences.

This course highlights some weak convergence techniques used in homogenization. The early theories tackled periodic structures — the theory of two-scale convergence [Nguetseng(1989), Allaire(1992)]. During the past decade and a half, there has been a new found interest in studying homogenization structures — as coined by G. Nguetseng in [Nguetseng(2003), Nguetseng(2004)]. In this new theory, the main objects of study are the so-called homogenization algebra — also referred to as algebra with mean value [Jikov et al.(1994)]. Using the Gelfand representation theory, the mathematical theory of classical periodic homogenization is extended to handle heterogeneous structures not necessarily periodic. These include for e.g. almost-periodic, quasi-periodic structures.

During the course, these new techniques will be applied to conductivity problems and to some simplified models of turbulent diffusion. The general plan of this course should include:

  1. Review of periodic homogenization: (a) Matched asymptotics [Bensoussan et al.(1978)], (b) Two-scale convergence method [Nguetseng(1989), Allaire(1992)], (c) Quantitative properties of effective coefficients.
  2. Σ–convergence [Nguetseng(2003), Nguetseng(2004)]: (a) Algebra with mean value, (b) Gelfand representation theory, (c) Homogenization of conductivity problem with almost-periodic coefficients.
  3. Turbulent diffusion [Pavliotis et al.(2008)]: (a) Diffusion enhancement — incompressible flows, (b) Diffusion depletion — compressible flows, (c) Competition between solenoidal and potential part of fluid fields.
  4. Convergence along mean flows [Holding et al.(2016)]: (a) Flows associated with vector fields, (b) Lagrangian stretching — growth of Jacobian matrix associated with flows, (c) Homogenization of advection-diffusion problem with fluid fields having multiple scales.
  5. Convergence in weighted spaces: (a) Polynomial growth in the Jacobian matrix associated with flows and its consequences on the homogenization problem, (b) Recovering some results from Freidlin-Wentzell theory [Freidlin et al.(1984)].
  6. Stochastic homogenization [Papanicolaou et al.(1978)]: (a) Strong advection problems with random solenoidal fluid fields, (b) Approximation of effective coefficients in stochastic setting [Bourgeat et al.(2004)].

Some of the topics covered in this course are increasingly active in the applied mathematics community. Lecture notes will be posted here.

Interested students will be given some easy-to-do analytical and numerical projects.

Pre-requisites:

It is preferable to have some basic knowledge of functional analysis.

For further information on the Taught Course Centre, click here.

References:

[Allaire(1992)] G. Allaire. Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, 23(6):1482–1518, 1992.

[Bensoussan et al.(1978)] A. Bensoussan, J-L. Lions, and G. Papanicolaou. Asymptotic methods in periodic structures. Stud. Math. Appl, 5, 1978.

[Bourgeat et al.(2004)] A. Bourgeat and A. Piatnitski. Approximations of effective coefficients in stochastic homogenization. Annales de l’IHP Probabilités et statistiques, 40:153–165, 2004.

[Freidlin et al.(1984)] M. Freidlin and A. Wentzell. Random perturbations. Random Perturbations of Dynamical Systems, pp:15–43, 1984.

[Holding et al.(2016)] T. Holding, H. Hutridurga, and J. Rauch. Convergence along mean flows. in press SIAM Journal on Mathematical Analysis, arXiv:1603.00424, 2016.

[Jikov et al.(1994)] V-V. Jikov, S-M. Kozlov, and O-A. Oleinik. Homogenization of Differential Equations and Integral Functionals. Springer-Verlag, Berlin, 1994.

[Nguetseng(1989)] G. Nguetseng. A general convergence result for a functional related to the theory of homogenization. SIAM Journal on Mathematical Analysis, 20(3):608–623, 1989.

[Nguetseng(2003)] G. Nguetseng. Homogenization structures and applications I. Zeitschrift für Analysis und ihre Anwendungen, 22(1):73–108, 2003.

[Nguetseng(2004)] G. Nguetseng. Homogenization structures and applications II. Zeitschrift für Analysis und ihre Anwendungen, 23(3):483–508, 2004.

[Papanicolaou et al.(1979)] GC. Papanicolaou and SRS. Varadhan. Boundary value problems with rapidly oscillating random coefficients. Random fields, 1:835–873, 1979.

[Pavliotis et al.(2008)] G. Pavliotis and A. Stuart. Multiscale methods: averaging and homogenization. Springer Science & Business Media, 2008