These are notes on Jean-Yves Chemin's Perfect Incompressible Fluids, in English translation. They cover: 1) All of Chapter 1; 2) Chapter 2 up through the end of the proof of Proposition 2.3.5 on p. 30, then starting again with the beginning of Section 2.4 on p. 32 and going to the end of Section 2.4 on the top of p. 37; 3) Section 3.1 of Chapter 3 and bits and pieces of the rest of the chapter; 4) Chapter 4 up to the middle of p. 68, then skipping to the top of page 72 and ending with the proof of Corollary 4.2.1 at the bottom of p. 78; Chapter 5 up to the end of Section 5.3 on p. 96; all of Chapter 6 (once I started on my "Vortex Sheets" lecture note, though, I stopped diligently updating the notes for this chapter).
Notes as of Saturday 18 October 2003 (with a comment on the proof of Theorem 5.1.1 added on 26 January 2006 and a comment on the last equation in Chapter 4 added on 19 June 2006): PS, PDF
Notes on Vortex Sheets, based primarily upon Chapter 6 of Chemin. This is essentially a very detailed rewrite of Chapter 6 of Chemin, and supersedes my notes, above, on that chapter. I created this document to prepare for a series of four lectures in the Fluid Mechanics seminar at UT given in April 2003. As of 31 May 2003, I have made the last changes I am planning to make for some time. The document is essentially complete, with any minor changes I eventually intend to make indicated by a "TODO" comment in the margin.
"Vortex Sheets" as of Friday 11 February (only significant change from 31 May 2003 is a rewrite of Appendix A on the spaces Em of Chemin, fixing a number of errors and hopefully clearing up a number of open issues): PS, PDF
This is a proof of the first part of Yudovich's theorem for unbounded vorticity. It can be seen as a detailed rewrite of Section 5.1 of Chemin's 1998 Perfect Incompressible Fluids (a translation of the 1995 French edition) for unbounded vorticity rather than bounded vorticity. The last major update was November 2003.
Small update 5 February 2005 (Remark added before Theorem 5.1). Also, I realized on 13 February that I had a ridiculously incorrect argument for the conservation of vorticity, removed the argument, and just added a comment on how to fix it.
PS, PDF
This is a proof of the second part of Yudovich's theorem for unbounded vorticity concerning the flow. It can be seen as a detailed rewrite of Section 5.2 of Chemin's 1998 Perfect Incompressible Fluids for unbounded vorticity rather than bounded vorticity. The results extending those of Section 5.2 on the modulus of continuity of the vector field and the flow are stated, in a slightly different form and without proof in Yudovich's 1995 paper in Mathematical Research Letters, Uniqueness Theorem for the Basic Nonstationary Problems in the Dynamics of an Ideal Incompressible Fluid.
Last update August 2003: PS, PDF
