Convener(s)
| Name: | Dr. Shirshendu Chowdhury | Dr. Rajib Dutta |
| Mailing Address: | Indian Institute of Science and Research - Kolkata, Mohanpur, Nadia – 741246, West Bengal, India |
Indian Institute of Science and Research - Kolkata, Mohanpur, Nadia – 741246, West Bengal, India |
| Email: | shirshendu at iiserkol.ac.in | rajib.dutta at iiserkol.ac.in |
This NCM workshop aims to introduce the participants to the mathematical analysis of incompressible Navier-Stokes equations, which is a basic model for the flow of incompressible viscous fluid. The main objective will be to study the existence and uniqueness of solutions (in the sense of weak, strong and mild) of incompressible Navier-Stokes equations via three methods: Galerkin approximation with energy methods, mollification approach and semigroup Theory with fixed Point methods based on regularity of initial conditions. We will compare the limitations and scope of these methods to study the different notions of solutions. These three methods have a wide range of applications in current research for studying the existence of solutions to other kinds of nonlinear partial differential
equations.
In the first week, we start with the detailed derivations of fluid flow equations. Then, introducing the divergence free spaces of hydrodynamics, we study the existence and uniqueness of solution (Weak/Strong) of Steady Stokes Equation as in the case of Laplace equation. Parallelly, we discuss the existence and uniqueness of solutions (weak/strong/mild) linear unsteady Stokes equations using Galerkin and Semigroup method, which is similar to the study of the heat equation.
In the second week, we will focus on existence and uniqueness of solution (weak/strong/mild)of the incompressible Navier-Stokes equations which is similar to semilinear heat equation. In this context, we will study local and global (with respect to time and initial data) well-posed-ness of Leray-Hopf weak solutions and strong solutions, mild solution for incompressible Navier-Stokes equations using above mentioned three methods and compare novelty and limitations of these methods.
At the final day of the workshop, we will conclude with the comments on unsolved open problems (The Millennium Prize Problems) in the three-dimensional case: the question of the global existence in time of regular/smooth solutions as well as the uniqueness of weak solutions, which inspires the large amount of active research in this direction.
Pre-requisites: The audience should be familiar with measure and integration, functional analysis, basic Sobolev spaces, and the well-posedness of ordinary differential equations. No background knowledge on fluid mechanics is required.
Target Audience: Ph.D. students, postdocs, scientists in research labs, young faculty members in educational institutions.
Dates:
Venue:
Venue Address:
Indian Institute of Science and Research-Kolkata, Mohanpur,
West Bengal-741246.
Venue State:
Venue City:
PIN:
Chrono Order:
Syllabus:
| Name of the Speakers with their affiliation. | No. of Lecture s | Detailed Syllabus |
| Dr. Debabrata Karmakar (TIFR-CAM) | 6 | Derivations of incompressible Navier-Stokes equa- tions and Euler equations: Inviscid and viscous fluids, Lagrangian and Eulerian descriptions of the flow, The Transport Theorem, Mass Conservation, Linear Momen- tum equation, Energy equations. Review of Basic Analysis tools: Weak, Weak^* conver- gence, L^p Spaces, Convolution, Hilbert Spaces, Spec- trum of Compact Self-Adjoint Operator, Sobolev Spaces, Sobolev inequalities and compactness theorems. Time dependent Sobolev spaces. |
| Dr. Mrinmay Biswas (IIT-Kanpur) |
6 | Linear Elliptic Equations: Existence and uniqueness of weak and strong solution of Laplace equation, Diver- gence free Spaces, Helmholtz decomposition, Leray pro- jection, De Rham’s theorem, existence and uniqueness of weak and strong Solution for steady Stokes equation, Stokes operator and its Spectral properties. |
| Dr. ShirshenduChowdhury (IISER-Kolkata) | 6 | Linear parabolic equations: Recall Key Results in Time dependent Sobolev spaces, Existence and uniqueness of weak/mild solutions via Galerkin/semigroup for Heat and unsteady Stokes Equa- tion. Regularity of weak solutions. Strong solutions for Heat and unsteady Stokes equation |
| Dr. Rajib Dutta (IISER-Kolkata) |
6 | Strong solutions for Navier-Stokes equations and Semilinear heat equation: Existence and uniqueness of local in time strong solutions in R^d, d=2,3 (and Torus) via Mollification with regular initial data , global (in time) existence in two dimension, small data global exis- tence in three dimension, Existence of local in time strong solutions in whole space for Euler Equation, |
| Prof. Ujjwal Koley (TIFR-CAM) |
6 | Weak solutions for Navier-Stokes equations and Semilinear heat equation: Existence of Leray-Hopf weak solutions for incompressible Navier-Stokes equa- tions in R^d, d=2,3 (and Torus)via Galerkin/Mollifica- tion, Uniqueness and regularity of weak solutions in d = 2, 3, Prodi-Serrin condition |
| Dr. Debayan Maity (TIFR-CAM) |
6 | Mild solutions for Navier-Stokes equations and Semi- linear heat equation: Existence and Uniqueness of Mild solutions, scaling criticality, Banach Fixed Point theo- rem(/Contraction Mapping Principle), local in time exis- tence and uniqueness for less regular initial data. Contin- uation of Solution. Comparison of Weak, Strong, Mild solution via 3 differ- ent methods : Galerkin, Mollification, Semigroup and Fixed Points. |
References:
- 1.J. Bedrossian and V. Vicol, The mathematical analysis of the incompressible Euler and Navier-Stokes equations. An introduction, vol. 225 of Grad. Stud. Math., Providence, RI:American Mathematical Society (AMS), 2022.
- F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier Stokes equations and related models, Applied Mathematical Sciences, 183. Springer, NewYork, 2013.
- R. Dautray, J-L Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5. Evolution problems. I. Springer-Verlag, Berlin, 1992.
- G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, 1994. Linearized steady problems.
- V. Girault, P-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin,1986
- P. G. Lemarie ́ -Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton,FL, 2002
- J. C Robinson, . Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.
- J. C. Robinson, J. L. Rodrigo, W. Sadowski, The three-dimensional Navier- Stokes equations, Classical theory, Cambridge Studies in Advanced Mathematics, vol. 157, CambridgeUniversity Press, Cambridge, 2016.
- 9. R. Temam, Navier-Stokes equations. Theory and numerical analysis. Revised edition. With an appendix by F. Thomasset, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam- New York.
Names of the tutors with their affiliation:
1. Keshav Sharma (TIFR CAM)
2. Manish Kumar (IISER K)
3. Sakil Ahmed (IISER K)
4. Debanjit Mandal (IISER K)
Time Table:
Week1
| Day | Date | Lecture 1 (9.30– 11.00) |
Tea (11.05– 11.25) |
Lecture 2 (11.30–1.00) |
Lunch (1.05– 1.55) |
Lecture 3 (2.00– 3.30) |
Tea (3.35- 3.55) |
Discussion (4.00- 5.00) |
Snacks 5.05- 5.35 |
| (name of the speaker) | (name of the speaker) | (name of the speaker) | (Name of the tutor) | ||||||
| Mon | 26/05/2 5 | DK | MB | SC | DK + SA+ DM | ||||
| Tues | 27/05/2 5 | DK | MB | SC | MB + SA+ DM | ||||
| Wed | 28/05/2 5 | DK | MB | SC | SC+ SA+ DM | ||||
| Thu | 29/05/2 5 | DK | MB | SC | DK+ SA+ DM | ||||
| Fri | 30/05/2 5 | DK | MB | SC | MB+ SA+ DM | ||||
| Sat | 31/05/2 5 | DK | MB | SC | SC+ SA+ DM |
Week 2
| Day | Date | Lecture 1 (9.30– 11.00) |
Tea (11.05– 11.25) |
Lecture 2 (11.30– 1.00) |
Lunch (1.05– 1.55) |
Lecture 3 (2.00– 3.30) |
Tea (3.35- 3.55) |
Discussion (4.00- 5.00) |
Snacks 5.05- 5.35 |
| (name of the speaker) | (name of the speaker) | (name of the speaker) | (Name of the tutor) | ||||||
| Mon | 02/06/25 | UK | RD | DM | UK + MK+KS | ||||
| Tues | 03/06/25 | UK | RD | DM | RD+ MK+KS | ||||
| Wed | 04/06/25 | UK | RD | DM | DM + MK+KS | ||||
| Thu | 05/06/25 | UK | RD | DM | UK + MK+KS | ||||
| Fri | 06/06/25 | UK | RD | DM | RD+ MK+KS | ||||
| Sat | 07/06/25 | UK | RD | DM | DM + MK+KS |
Full forms for the abbreviations of speakers and tutors:
Speakers :
DK - Debabrata Karmakar, RD - Rajib Dutta, SC - Shirshendu Chowdhury, UK- Ujjwal Koley, MB -Mrinmay Biswas, DM - Debayan Maity.
Tutors:
KS - Keshav Sharma, MK- Manish Kumar , SA - Sakil Ahmed, DM-Debanjit Mandal
Selected Applicants:
| Sr.n | SID | Full Name | Gender | Affiliation | Position in College/ University | University/ Institute M.Sc./ M.A. | Year of Passing M.Sc./ M.A | Ph.D. Degree Date |
| 1 | 61736 | Mr. Haradhan Dutta | Male | IIT Kharagpur | PhD | NIT Rourkela | 2022 | |
| 2 | 61747 | Mr. Sandip Roy | Male | IIT Indore | Ph.D Student | University of North Bengal | 2022 | |
| 3 | 61776 | Mr. Ashish Jain | Male | IIT Indore | Phd student | Doctor Harisingh gour central university sagar | 2022 | |
| 4 | 61794 | Dr. Alok Kumar Sahoo | Male | Bhadrak Autonomous College | Asst. Prof. | Pondicherry University | 2017 | 06/01/2023 |
| 5 | 61875 | Mr. Swarupananda Banerjee | Male | Indian Institute of Technology Kharagpur | PhD | Ramakrishna Mission Vivekananda Educational and Research Institute | 2022 | |
| 6 | 61881 | Ms Anupama Ghorai | Female | National Institute of Technology, Tiruchirappalli | Research Scholar | Midnapore College (Autonomous) | 2019 | |
| 7 | 61966 | Mr Aman Patel | Male | Indian Institute of Technology Kanpur | PhD Student | Indian Institute of Technology Kanpur | 2018 | |
| 8 | 62112 | Mr. Ram Gopal Jaiswal | Male | IIT Roorkee | PhD | IIT Roorkee | 2019 | |
| 9 | 62114 | Mr. Saroj Si | Male | IIT Roorkee | PhD | Utkal University, Vanivihar, Bhubaneswar | 2018 | |
| 10 | 62172 | Dr. Wasim Akram | Male | IIT Roorkee | NBHM Postdoctoral Fellow | IIT Bombay | 2017 | 19/08/2023 |
| 11 | 62320 | Mr. Akram Khan | Male | Indian institute of technology Roorkee | phD | Indian institute of technology Kharagpur | 2024 | |
| 12 | 62450 | Mr. Pankaj Roy | Male | IIT Guwahati | PhD | Indian Institute of Technology Kanpur | 2022 | |
| 13 | 62562 | Mr. Saptarshi Halder | Male | Indian Statistical Institute, Bangalore, India | M.Math Student | Indian Statistical Institute, Bangalore | Appeared | |
| 14 | 62628 | Mr. Mohmedmunavvar Mubarak Bapu | Male | Indian Institute of Technology Kanpur | Ph.D | 2018 | ||
| 15 | 62744 | Mr. Krishnagopal Maitra | Male | IIT KANPUR | PhD scholar | University of Kalyani | 2022 | |
| 16 | 62819 | Mr. Ankit Dhara | Male | Indian Institute of Technology Bhubaneswar | PhD Student | Indian Institute of Technology Bhubaneswar | 2024 | |
| 17 | 62826 | Ms Saheli Panja | Female | National Institute of Science Education and Research | Integrated PhD | National Institute of Science Education and Research | ||
| 18 | 63179 | Mr. Vivek Sahu | Male | IIT Kanpur | Ph.D. | IIT Guwahati | 2020 | |
| 19 | 63301 | Ms Manisa Maity | Female | IISER Bhopal | PhD Student | IIT Tirupati | 2023 | |
| 20 | 63314 | Mr. Ashutosh Dixit | Male | IIT Jodhpur | PhD | Dr. Hari Singh Gour Vishwavidyalaya (A Central University) Sagar | 2022 | |
| 21 | 63325 | Mr. Harsh Tiwari | Male | Indian Institute of Technology Roorkee | PhD | Banaras Hindu University | 2024 | |
| 22 | 63380 | Mr Madepalli Sekharu | Male | Indian Institute Of Technology | PhD | M.sc | 2020 | |
| 23 | 63381 | Ms. Stuti Das | Female | Indian Institute of Technology Kanpur | PhD | Jadavpur University | 2021 | |
| 24 | 63503 | Dr. Pardeep Kumar | Male | National Institute of Science Education and Research (NISER), Bhubaneshwar | Post Doctoral Fellow | South Asian University, New Delhi | 2015 | 13/12/2024 |
| 25 | 63662 | Mr. Om Tripathi | Male | Indian Institute Of Technology Roorkee Saharanpur Campus | PhD | National Institute Of Technology Warangal | 2023 | |
| 26 | 63728 | Mr. Surajit Pal | Male | IIT Bombay | PhD | IIT Kanpur | 2022 | 27/07/2023 |
| 27 | 63948 | Mr. Daniel Vijay L | Male | Indian Institute of Technology Goa | PhD | Bharathidasan University | 2020 | |
| 28 | 64034 | Mr. Aman Batra | Male | IIT Bombay | PhD | IIT Kharagpur | 2019 | |
| 29 | 64035 | Mr. Anurag Tiwari | Male | TIFR | Integrated PhD student | TIFR Bengaluru | ||
| 30 | 64075 | Ms. Rinki Rawat | Female | Indian Institute of Technology Kanpur | PhD | University of Lucknow | 2019 | |
| Local participants | ||||||||
| 1 | 61745 | Ms Shweta Mishra | Female | Maulana Abul Kalam Azad University of Technology, West Bengal, India | Assistant Professor | Banaras Hindu University, Varanasi, UP, India | 2008 | 14/01/2025 |
| 2 | 62076 | Mr. Debraj Kar | Male | University of Kalyani | PhD | University of Kalyani | 2020 | |
| 3 | 62077 | Mr. Biren Karmakar | Male | University of Kalyani | Phd | KAZI NAZRUL UNIVERSITY | 2018 | |
| 4 | 62100 | Mr. Sk Jishan Ali | Male | Indian institute of Science Education and Research Kolkata | MSc Student | Indian institute of Science Education and Research Kolkata | ||
| 5 | 62222 | Mr Debasish Paul | Male | Indian Institute of Science Education and Research Kolkata | Msc Student | IISER KOLKATA | 2025 | |
| 6 | 62367 | Mr Kishan Saha | Male | IISER KOLKATA | BS MS 4th year Student | IISER KOLKATA | Appeared | |
| 7 | 62464 | Mr. Utsav Dewan | Male | Indian Statistical Institute, Kolkata | PhD student | Indian Statistical Institute, Kolkata | 2020 | |
| 8 | 63096 | Ms. Susmita Saha | Female | University of Calcutta | PhD Student | University of Calcutta | 2018 | |
| 9 | 63263 | Ms Deepta Roy | Female | Indian Institute of Science Education and Research Kolkata | MSc Student | Indian Institute of Science Education and Research Kolkata | Appeared | |
| 10 | 63768 | Mr. Gourav Banerjee | Male | IISER KOLKATA | Student | |||
How to Reach:
TBA