An introduction to incompressible Navier-Stokes equations
1. Temam, Roger Navier-Stokes equations. Theory and numerical analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI,2001.
2. Seregin, Gregory Lecture notes on regularity theory for the Navier-Stokes equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
3. Galdi, G. P. An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011.
640:517 or permission from instructor
In this course we will introduce some results on incompressible Navier-Stokes equations.
The material being presented will be as follows.
For some parts, detailed proofs will be given, while for some other parts outlines of proofs will be given.
One main objective is to discuss interesting open problems in stationary incompressible Navier-Stokes equations (some of them are perhaps doable)
1. Survey and exposition on the existence of solutions to the nonhomogeneous stationary incompressible Navier-Stokes equations in dimension two. This will includes works from some pioneering work of Leray and new developments in the last few years, which includes in particular the solution of the problem in two dimension:
[ Korobkov, Mikhail V.; Pileckas, Konstantin; Russo, Remigio, Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains. Ann. of Math. (2) 181 (2015), no. 2, 769–807.] The problem is dimension three remains open.
2. Exposition on some works on Leray's problem of steady Navier-Stokes flow past a body in the plane. A long standing challenging open problem in this direction will be mentioned.
3. Regularity of solutions of stationary incompressible Navier-Stokes equations in dimension less than 5, partial regularity of stationary Navier-Stokes equations in dimension bigger than 4, existence of regular solution of the stationary Navier-Stokes equations in dimension 5.
4. Existence of weak solutions in three space dimension (Jean Leray) and the existence and smoothness of Navier-Stokes solutions in two space dimension (Ladyzhenskaya).
5. Global existence of solution for Navier-Stokes if the initial data is small in various scale invariant spaces.
6. Partial regularity of weak solutions in three space dimension (Caffarelli-Kohn-Nirenberg, and a simplified proof by Fanghua Lin).
7. L^\infty([0,T], L^3(R^3)) solutions of the Navier-Stokes and backward uniqueness (Escauriaza-Seregin-Sverak).