NCMW - Theoretical aspects of Incompressible Navier-Stokes equations (2025)
Venue: Indian Institute of Science and Research, Kolkata
Dates: 26 May 2025 to 7 Jun 2025
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.