Advanced topics in PDEs (2023)

Convener(s)

 
Name: Prof. Ujjwal Koley,
Associate Professor		 Prof. Rakesh Kumar Bajaj,
 Professor & HOD		 Dr. Bidhan Chandra Sardar,
Assistant Professor
Mailing Address: TIFR Centre for Applicable Mathematics
Sharada Nagar, Chikkabommasandra,
Bengaluru, 560065		 Department of Mathematics, Jaypee University of Information Technology
Waknaghat, Solan, H.P - 173234		 Department of Mathematics
Indian Institute of Technology Ropar
Rupnagar, Punjab - 140001
Email: ujjwal at math.tifrbng.res.in  rakesh.bajaj at juitsolan.in bcsardar at iitrpr.ac.in

This NCMW aims to introduce the participants to the advanced topics in Partial Differential Equations (PDEs). This will be indeed immensely helpful to the researchers who are already familiar with the basics of PDEs, since we are going to discuss several state-of-the art techniques related to Hyperbolic, Elliptic and Parabolic PDEs.

In the first week, we plan to discuss topics like BV compactness for scalar conservation laws, controllability of PDEs with their applications, and homogenization of elliptic PDEs.

At the end of the first week, we plan to focus more on compensated compactness method for scalar conservation laws, Caffarelli-Silvestre extension problems for fractional PDEs, and concentration compactness principle and their applications in the Sobolev embedding.

Dates: 

Monday, May 15, 2023 - 19:00 to Saturday, May 27, 2023 - 19:00

Venue: 

Venue Address: 

Waknaghat, Solan, Himachal Pradesh, 173234.

Venue State: 

Venue City: 

PIN: 

173234

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)

Selected Applicants: 

 

Sr.n SID Full Name Gender University/ Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Deg. date
1 46370 Mr. Ashish Ashish Male IIT Delhi 2022  
2 46455 Mr. Vivek Sahu Male IIT Guwahati 2020  
3 46474 Ms. Neelam Yadav Female M.sc 2020  
4 46478 Ms. Jyoti Jindal Female Punjabi University, Patiala 2020  
5 46479 Mr. Ayush Agrawal Male IIT Gandhinagar 2019  
6 46490 Mr. Atanu Manna Male Indian Institute of Technology, Hyderabad 2021  
7 46509 Mr. Rohit Kumar Male University of Lucknow 2018  
8 46522 Mr Mukul Dwivedi Male IIT Guwahati 2020  
9 46602 Mr. Sanjit Biswas Male Indian institute of technology Kanpur. 2020  
10 46629 Ms. Stuti Das Female Jadavpur University 2021  
11 46663 Ms. Anamika Pandey Female IIT Kharagpur 2021  
12 46698 Mr Dattatreya N N Male Mangalore University 2019  
13 46765 Ms. Anusree R Female Pondicherry University 2018  
14 46807 Ms. Anupma Arora Female Guru Jambheshwar University of Science & Technology Hisar 2019  
15 46811 Mr. Mohmedmunavvar Mubarak Bapu Male SPU 2018  
16 46929 Ms. Aditi Chattaraj Female Bankura University 2019  
17 46951 Ms. Mansi Singh Female Indian Institute of Technology Kharagpur 2019  
18 47009 Mr. Lovelesh Sharma Male Central University of South Bihar, Patna 2018  
19 47047 Ms. Subhashree Sahu Female Ravenshaw University 2020  
20 47060 Ms. Anamika Purohit Female The Maharaja Sayajirao University of Baroda 2019  
21 47064 Mr. Ravindra Singh Chauhan Male IIT Bhubaneswar 2019  
22 47077 Mr. Deepak Kumar Mahanta Male Dr B.R.Ambedkar National Institute of Technology Jalandhar 2019  
23 47126 Mr. Anirban Das Male TIFR - Centre for Applicable Mathematics Appeared  
24 47136 Mr. Sahil Kundu Male TIFR CAM 2018  
25 47158 Mr. Soumak Nag Male IIEST Shibpur 2019  
26 47160 Mr. Paramananda Das Male IISER Pune Appeared  
27 47165 Mr. Pranav Kumar Male IISER BHOPAL 2022 25/11/2022
28 47225 Mr. Haradhan Dutta Male M. Sc. 2022  
29 47227 Mr. Jhantu Pal Male M.Sc. 2021  
30 47253 Ms Asmita Rai Female Udai Pratap Autonomous College 2019  
31 47259 Mr. Debraj Kar Male University of Kalyani 2020  
32 47320 Mr. Debanjit Mondal Male IISER KOLKATA 2022  
33 47321 Mr. Manish Kumar Male Indian Institute of Science Education and Research, Kolkata 2021  
34 47331 Ms. Rupali Kumari Female National Institute of Technology Jamshedpur 2018  
35 47359 Mr. Keshav Sharma Male Center for Applicable Mathematics, TIFR Appeared  
36 47368 Mr. Vikas Jaiswal Male IIT Bhubaneswar 2020  
37 47376 Mr. Sumit Maity Male Vidyasagar University 2019  
38 47537 Mr. Shammi Malhotra Male IIT Delhi 2022  
39 47576 Mr. Anupam Nayak Male TIFR CAM    
40 47597 Ms. Lakshmi R Female University Of Calicut 2022  
41 47693 Mr. Tuhin Mondal Male IIT Madras 2022  
42 47979 Mr. Pradipta Chatterjee Male TIFR CAM Appeared  
43 48104 Ms. Suman Kanungo Female Pt. Ravishankar Shukla University, Raipur (Chhattisgarh) 2020  
44 48118 Ms. Harsita Srivastava Female Lucknow University 2018  
45 48143 Ms. Priyanka . Female M. Sc. 2019  
46 48179 Ms. Poonam Rani Female Indian Institute of Technology, Kanpur ( IIT-KANPUR) 2021  

 

 

How to Reach: 

How to reach:

JUIT Waknaghat is midway between Shimla & Solan . Shimla is 24 Kms from the campus. Kalka is the nearest broadgauge railway station. Waknaghat is very well connected with buses from Delhi, UP, Haryana, Punjab, Rajasthan and Chandigarh/Kalka.

Waknaghat is 24 kms. from Solan on the way to Shimla on NH-22. For coming to JUIT, one has to take a left turn at Waknaghat and drive for 3 kms. From JUIT, Shimla is 24 kms. The location is at a height of 1550 m above MSL.

TEMPERATURE – Max. 35°C Min. – 2°C

CLOTHING: Cotton clothes in Summer and heavy woolens in winter.

LANGUAGE: Hindi, English are understood and spoken by the people .

About Himachal

Surrounded by the purple-blue hue on the mountains and spread across seven hills in the Himalayas with steep forested hillsides, long winding roads, bustling bazaars and a glorious past, Shimla – The capital city of Himachal Pradesh is spell binding. Visit Shimla any time of the year and the historic hilly town is never shy of happy vacationers. Not only known for its landscapic beauty, the city has been growing into an educational hub. From prestigious schools like Bishop Cotton and Auckland House, to colleges like HPU and St. Bede’s, the city speaks volumes of the knowledge it has stored. Jaypee University of Information Technology (JUIT), centred between the two major cities, Shimla and Solan has grown profoundly in the past decade. The serenity, the calmness of the environment gives an internal boost and pump to your learning skills. All you need to do is, feel the air, gaze upon the lush landscapes and sing the song of your life. Also when the time comes to bid adieu to this place with heaps of knowledge, tonnes of happy memories and a heavy heart.

How to reach JUIT:

By Road:

There is good connectivity of the campus with major cities around. It lies three kilometres away from the national highway. HRTC, Haryana and Punjab Roadways buses along with private travellers are easily available from Shimla as well as Chandigarh. They drop you off at Waknaghat from where cab service till the campus are available 24*7 all around the year.

By Train:

The nearest broad gauge railway station is Chandigarh or Kalka from where buses, and taxis are available. It takes approx 2 hours by road to Waknaghat

By Air:

The convenient all season airport is Chandigarh from where buses, taxis are available to Waknaghat . It takes approx 2.5 hours to JUIT.

 

 

School Short Name: 

atpdes

Last Date Application: 

Saturday, April 15, 2023

School Type: 

NCMW

Separate faculty form: 

0