Summer Graduate School

Fluid-flow stream function color-coded by vorticity in 3D flat torus calculated by K. Nakai (The University of Tokyo)

The purpose of the workshop is to introduce graduate students to fundamental results on the Navier-Stokes and the Euler equations, with special emphasis on the solvability of its initial value problem with rough initial data as well as the large time behavior of a solution. These topics have long research history. However, recent studies clarify the problems from a broad point of view, not only from analysis but also from detailed studies of orbit of the flow.

General prerequisites:

• H. Brezis, 
  Functional analysis, Sobolev spaces and partial differential equations. 
  Springer, New York, 2011. 
  ISBN: 978-0-387-70913-0
• L. C. Evans, 
  Partial differential equations. Second edition. 
  Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. 
  ISBN: 978-0-8218-4974-3

Prerequisites by lecturer:

1. Lectures by L. Brandolese
a) J.-Y. Chemin, 
Localization in Fourier space and Navier-Stokes system. 
Lectures notes of the De Giorgi Center, 2005. 
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwj8lPnh4rncAhVGrxoKHQYQBccQFggvMAA&url=https%3A%2F%2Fwww.math.uzh.ch%2Fpde13%2Ffileadmin%2Fpde13%2Fpdf%2FcoursPisa.pdf&usg=AOvVaw1RorDZAuT8VSrh-Y7jX0Om
Phase space analysis of partial differential equations. Vol. I, 53–135, 
Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004. 
In particular, Chapter 1 and Sections 2.1, 2.2, 2.3.

b) A. J. Chorin and J. E. Marsden, 
A mathematical introduction to fluid mechanics. Third edition. 
Springer, 　Corrected fourth printing 2000.
Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1993. 
  ISBN: 0-387-97918-2 
Soft cover :ISBN: 978-1-4612-6934-2
In particular, Chapter 1

2. Lectures by I. Gallagher
a) H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. 
Springer, New York, 2011. 
ISBN: 978-0-387-70913-0
In particular, Chapters 4, 5, 6, 8, 10

b) H. Bahouri, J.-Y. Chemin and R. Danchin, 
Fourier analysis and nonlinear partial differential equations. 
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011. 
ISBN: 978-3-642-16829-1
In particular, Chapter 1 and Section 3.1

c) L. C. Evans, Partial Differential Equations.  Second edition. 
Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. 
ISBN: 978-0-8218-4974-3
In particular, Chapters 1, 2, 3, 5

3. Lectures by T. Yoneda
a) A. Majda and A. Bertozzi,
Vorticity and Incompressible Flow,
Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge 2002. 
ISBN: 0-521-63057-6; 0-521-63948-4 
In particular, Chapter 4

b) J. Bourgain and D. Li,
Strong ill-posedness of the incompressible Euler equations in borderline Sobolev spaces. 
Invent. math. 201, (2015), 97-157.
 

For eligibility and how to apply, see the Summer Graduate Schools homepage