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. 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.
  2. 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.
  3. R. Dautray, J-L Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5. Evolution problems. I. Springer-Verlag, Berlin, 1992.
  4. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, 1994. Linearized steady problems.
  5. 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
  6. 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
  7. 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.
  8. 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. 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

 

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