Brief description:

Navier-Stokes equations is a basic mathematical model to describe motion of a viscous incompressible fluid. In the 1930s Jean Leray proved existence of a weak solution defined globally in time. Uniqueness of weak solutions in 3D remains an open question. At the beginning we will cover the fundamentals of the theory including results of Leray and contributions of the later authors such as E.Hopf, O.Ladyzhenskaya, J.-L.Lions, G.Prodi, and others. In the remaining time we will concentrate on some of the striking recent advances.

1. Variational formulation of the Navier-Stokes equations. Weak solutions.

2. Uniqueness in dimension 2.

3. Nonexistence of self-similar blow-up for 3D Navier-Stokes equation.

4. The theory of T.Kato.

5. Littlewood-Paley decomposition and paraproducts.

6. Function spaces (Besov, Morrey-Campanato, Lorentz,...).

7. Uniqueness of mild solutions.

Prerequisite: Functional analysis as in the Applied Math prelim course or equivalent. PDE's as in the Introduction to PDE's course.

Textbooks: None required.

References:

1. J.Leray, \OEuvres scientifiques, vol.2, Springer and Soc. Math. France, 1998.

2. O.A.Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, 1969.

3. J.-L.Lions, Quelques m\'ethods de r\'esolution de probl\`emes aux limites nonlin\'eaires, Dunod, 1969.

4. R.Temam, Navier-Stokes equations, North Holland, 1984.

5. M.Cannone, Ondelettes, paraproduits et Navier-Stokes, Diderot, 1995.

6. Y.Meyer, Wavelets, paraproducts and Navier-Stokes equations.

7. P.-L.Lions, Mathematical topics in fluid mechanics, vol.1, Oxford Univ. Press, 1996

Consent of Instructor Required?   No.
 

Prof. Mikhail Vishik

RLM 11.128

471-1165

Email: vishik@math.utexas.edu