Yanyan Li
Subtitle:
An introduction to incompressible NavierStokes equations
Text:
1. Temam, Roger NavierStokes 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 NavierStokes equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
3. Galdi, G. P. An introduction to the mathematical theory of the NavierStokes equations. Steadystate problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011.
Prerequisites:
640:517 or permission from instructor
Course Description:
In this course we will introduce some results on incompressible NavierStokes 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 NavierStokes equations (some of them are perhaps doable) 1. Survey and exposition on the existence of solutions to the nonhomogeneous stationary incompressible NavierStokes 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 NavierStokes 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 NavierStokes 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 NavierStokes equations in dimension less than 5, partial regularity of stationary NavierStokes equations in dimension bigger than 4, existence of regular solution of the stationary NavierStokes equations in dimension 5. 4. Existence of weak solutions in three space dimension (Jean Leray) and the existence and smoothness of NavierStokes solutions in two space dimension (Ladyzhenskaya). 5. Global existence of solution for NavierStokes if the initial data is small in various scale invariant spaces. 6. Partial regularity of weak solutions in three space dimension (CaffarelliKohnNirenberg, and a simplified proof by Fanghua Lin). 7. L^\infty([0,T], L^3(R^3)) solutions of the NavierStokes and backward uniqueness (EscauriazaSereginSverak). 
