If not me, who? And if not now, when?
by Mikhail Gorbachev

Notes on INS (16): Random thoughts

Infinite speed is no big deal ?

There may be some very small regions in space-time where there are some points with infinite speed. The flow will sill be continuous. From the “physical” point of view, the flow map is more important than the velocity field ? The flow maps may have a few . . . → Read More: Notes on INS (16): Random thoughts

Cơ chế đánh giá khoa học mới của Đức

Dưới đây là bài viết của Phùng Hổ Hải trao đổi trên mailing list của các nhà khoa học VN, tôi xin phép đăng lại. Một điểm rất đáng chú ý là, họ rất quan tâm đến chất lượng, thay vì số lượng như ở VN. Cái trò “băm nhỏ”, công bố các kết . . . → Read More: Cơ chế đánh giá khoa học mới của Đức

Notes on INS (15): vulgarization article by Isabelle Gallagher (in French)

http://images.math.cnrs.fr/Autour-des-equations-de-Navier.html

This article (in French) by Gallagher, written for a large public, contains many interesting tidbits around the Navier-Stokes equations, e.g. history of the letter nabla (, which is NOT a Greek letter).

Je fais ici un copier-coller:

Autour des équations de Navier-Stokes Le 28 janvier 2010, par Isabelle Gallagher Professeur à l’Institut de . . . → Read More: Notes on INS (15): vulgarization article by Isabelle Gallagher (in French)

Entropy of geometric structures

Last updated: 22/Oct/2010

(This is the first draft of a note to be submitted to a special issue of the Bulletin of the Brazilian Mathematical Society)

Geometric structures

In this note, by a geometric structure, we mean a normed vector bundle $A \to M$ over a vector bundle (i.e. on each fiber there is . . . → Read More: Entropy of geometric structures

Notes on INS (14): CKN theory

Last updated: 27/Oct/2010

CKN stands for Caffarelli-Kohn-Nirenberg. The theory is about partial regularity of solutions of INS. One should probably add the name of Scheffer, who introduced the concepts that CKN improved/generalized.

The main result is that the (parabolic) 1-dimensional Hausdorff measure of the (hypothetical) singular set in space-time is zero (which means that, . . . → Read More: Notes on INS (14): CKN theory

Notes on INS (13): Sinai’s papers

This part is about Yakov Sinai‘s papers on INS. Sinai wrote I-don’t-know-how-many papers on the Navier-Stokes equations. It seems that he has been trying for many years to study the properties of the N-S equation from many different points of view, and has a lot of interesting ideas, some of which may turn out . . . → Read More: Notes on INS (13): Sinai’s papers

Notes on INS (11): Serrin’s regularity result

Last updated: 20/Oct/2010

In this part we will prove the following theorem of Serrin (1962):

Serrin’s Theorem. If and is a weak solution such that

for some region of space-time, then for .

The exposition will mainly follow lecture notes by James Robinson (Campinas, 2010), Chapter II. (These lecture notes are much . . . → Read More: Notes on INS (11): Serrin’s regularity result

Notes on INS (12): Lagrangian picture

Lagrangian picture

- Uniqueness of trajectories

Refs: Robinson, Chemin, …

(to be added)

Notes on INS (10): Serrin-type regularity criteria based on pressure

Last updated: 17/Oct/2010

Serrin-type regularity criteria based on pressure

Serrin was the first to write down a criterion for global smoothness of solutions of the INS equation in terms of some a-priori inequalities. In the 2D case, his inequalities are automatically satisfied, and so he proved regularity for 2D INS. In this part, we . . . → Read More: Notes on INS (10): Serrin-type regularity criteria based on pressure

Notes on INS (9): Ladyzhenskaya’s work on Navier-Stokes

Last updated: 19/Oct/2010

In this part: Ladyzhenskaya’s contributions to INS

Olga Ladyzhenskaya (1922-2004) is famous for her work on PDEs (regularity of elliptic and parabolic equations etc.), and especially the Navier-Stokes problem. In 2002 she was awarded the “Doctor Honoris Causa” by the University of Bonn. There is an interesting write-up by  Michael Struwe . . . → Read More: Notes on INS (9): Ladyzhenskaya’s work on Navier-Stokes

Giải Ếch Vàng cho bộ sách Chào Lớp 1 của Cánh Buồm ?

Giải Ếch Vàng (Yellow Frog Prize) là một trong những giải thưởng danh giá nhất thế giới, ghi nhận những thành tích nổi bật nhất trên thế giới trong các lĩnh vực khác nhau. Giải được Viện Hàn Lâm Thành Tích Vĩ Đại Toàn Cầu (World Academy of Greatest Achievements) xét và trao tặng. . . . → Read More: Giải Ếch Vàng cho bộ sách Chào Lớp 1 của Cánh Buồm ?

Notes on INS (8): Additional results and references (unsorted)

A list of other references and results (in no particular order)

1) A. Vasseur, Regularity criterion for 3d navier-stokes equations in terms of the direction of the velocity, Applications of Mathematics,Volume 54 (2009), Number 1, 47-52.

Abstract: In this short note, we give a link between the regularity of the solution u to the . . . → Read More: Notes on INS (8): Additional results and references (unsorted)

Notes on INS (7): Video lectures on Navier-Stokes

Video lectures about INS

Lecture by Grigory Seregin,

St Petersbourg, 17/sep/2010 (very interesting talk in Russian, about regularity of solutions):

http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=1296

In the 2-dimensional case, there is a multiplicative inequality of Ladyzhenskaya, which leads to the global regularity of solutions.

Lecture by Luis Caffarelli (elementary explanation of the problem, Clay Math Institute — . . . → Read More: Notes on INS (7): Video lectures on Navier-Stokes