NCMW - Theoretical aspects of Incompressible Navier-Stokes equations (2025)
Speakers and 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