NCMW - Advanced topics in PDEs (2023)

Speakers and Syllabus

 Syllabus to be covered in terms of modules of 6 lectures each:

Name of the Speakers with their affiliation.

No. of Lectures

Detailed Syllabus

Prof. Ujjwal Koley
(TIFR-CAM)

6

Laplace equation, Fundamental solution for Laplace equation, mean value properties, maximum principle, Regularity of solutions to Laplace equation, Harnack inequality, Green function, Energy method for Laplace equation.

Dr. Debayan Maity
(TIFR-CAM)

6

Controllability of linear control systems: various notions of controllability, Duality between controllability and observability, controllability of transport equation.

Dr. Rajib Dutta
(IISER-Kolkata)

6

Weak solution for scalar conservation laws, Solution of Riemann problems, Vanishing viscosity method, Bounded variation (BV) compactness and its limitations.

Dr. Ali Hyder
(TIFR-CAM)

6

Introduction to fractional Laplacian, Definition via Fourier transform and singular integral, basic properties, Fundamental solution, maximum principle, Caffarelli-Silvestre extension.

Dr. Arka Mallick
(IISC)

6

Concentration Compactness principle and their applications in the Sobolev embedding.

Dr. Bidhan Sardar
(IIT-Ropar)

6

Homogenization of Elliptic Partial Differential Equations: Rapidly oscillating periodic function, Weak limits of rapidly oscillating periodic functions, Two-scale convergence method.

 

References:

  1. Alberto Bressan,  Hyperbolic System of Conservation laws. The One Dimensional Cauchy Problem. Oxford Lecture Series in Mathematics and its Applications,20. Oxford University Press, Oxford, 2000.

  2. C. Dafermos, Hyberbolic Conservation Laws in Continuum Physics, Springer-Verlag, Heidelberg (2000).

  3. D. Cioranescu, and P. Donato, An Introduction to homogenisation, Oxford university press, 1999.

  4. E. Godlewski and P. A. Raviart,  Hyperbolic System of Conservation Laws, Collection Mathematiques et Applications de las SMAI, Ellipses, Paris (1991).

  5. L.C. Evans, Partial Differetial Equations, Graduate Studies in Mathematics, Vol 19, AMS 1998.

  6. H. Holden and N. H. Risebro, Front Tracking for Hyperbolic System of Conservation Laws, Springer, 2015.

  7. Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator.

  8. Luis Caffarelli, and Luis Silvestre, An extension problem related to the fractional Laplacian.

  9. E. D. Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces.

  10. Jean-Michel Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, AMS , 136, 2007.

  11. Sorin Micu and Enrique Zuazua, An Introduction to the Controllability of Partial Differential Equations. Available at http://www.uam.es/personal_pdi/ciencias/ezuazua/informweb/argel.pdf.

  12. Marius Tucsnak and George Weiss, Observation and Control for Operator Semigroups, 494 pages, BirkhäuserBasel-Boston-Berlin, 2009.

  13. Struwe, Michael(1996), Variational methods. Applications to nonlinear PDEs and Hamiltonian systems, Second edition. Ergebnisse der Mathemati und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34. Springer Verlag, Berlin.

  14. J. Malek et al., Weak and measure-valued solutions to evolutionary PDEs, Applied mathematics and mathematical computation 13, 1996.

 

Time Table

Tentative time-table, mentioning names of the speakers and tutors with their affiliation:

WEEK I

Day

Date

Lecture 1

(9.30
to
11.00)

Tea

(11.05
to
11.25)

Lecture 2

(11.30
to
1.00)

Lunch

(1.05
to
1.55)

Lecture 3

(2.00
to
3.30)

Tea

(3.35
to
3.55)

Discussion

(4.00
to
5.00)

Snacks

5.05
to
5.35

 

 

(name of the speaker)

 

(name of the speaker)

 

(name of the speaker)

 

(Name of the tutor)

 

Mon

15th May, 2023

DM

 

BS

 

RD

 

Abu + DM

 

Tues

16th May, 2023

DM

 

BS

 

RD

 

Abu + DM

 

Wed

17th May, 2023

DM

 

BS

 

RD

 

Abu + BS

 

Thu

18th May, 2023

DM

 

BS

 

RD

 

Abu + BS

 

Fri

19th May, 2023

DM

 

BS

 

RD

 

Vaibhav + RD

 

Sat

20th May, 2023

DM

 

BS

 

RD

 

Vaibhav + RD

 

 

WEEK II

Day

Date

Lecture 1

(9.30
to
11.00)

Tea

(11.05
to
11.25)

Lecture 2

(11.30
to
1.00)

Lunch

(1.05
to
1.55)

Lecture 3

(2.00
to
3.30)

Tea

(3.35
to
3.55)

Discussion

(4.00
to
5.00)

Snacks

5.05
to
5.35

 

 

(name of the speaker)

 

(name of the speaker)

 

(name of the speaker)

 

(Name of the tutor)

 

Mon

22th May, 2023

UK

 

AH

 

AM

 

Vaibhav + UK

 

Tues

23th May, 2023

UK

 

AH

 

AM

 

Vaibhav + UK

 

Wed

24th May, 2023

UK

 

AH

 

AM

 

Ritu + AH

 

Thu

25th May, 2023

UK

 

AH

 

AM

 

Ritu + AH

 

Fri

26th May, 2023

UK

 

AH

 

AM

 

Ritu + AM

 

Sat

27th May, 2023

UK

 

AH

 

AM

 

Ritu + AM

  

UK: Ujjwal Koley || BS: Bidhan Sardar || RD: Rajib Dutta || DM: Debayan Maity || AH: Ali Hyder || AM: Arka Mallick || AS: Abu Sufian || VJ: Vaibhav Kumar Jena || RR: Ritu Raj

Names of the tutors with their affiliation :

    1. Dr. Abu Sufian (TIFR-CAM),
    2. Dr. Vaibhav Kumar Jena (TIFR-CAM)
    3. Mr. Ritu Raj (IIT Ropar)

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