Toute méchanceté vient de faiblesse.
by Jean-Jacques Rousseau, Émile.

Notes on INS (6): Failed attempts to prove regularity

updated 17/Oct/2010

False proofs of regularity ?

The difficulty of a problem can be measured by the number of false proofs (by professional mathematicians, not counting the amateur ones) it receives. Since the regularity problem of the INS equation is notoriously difficult, it must surely attract quite a few false proofs. I’ll collect them . . . → Read More: Notes on INS (6): Failed attempts to prove regularity

Notes on INS (5): L10/3 and L5/3 regularity

Last updated: 28/Oct/2010

This part:

- L5/3 regularity of pressure (Sohr-von Wahl theorem)

- L10/3 regularity of velocity field.

References:

- Sohr & von Wahl

- Robinson’s lecture notes

- … ?

(to be added)

Phân tích kỹ thuật (1)

Last updated: 02/Nov/2010

Seminar toán tài chính ở Toulouse (hiện tại có mười mấy người tham dự, chủ yếu là NCS hay sinh viên cao học về toán) bắt đầu thảo luận về phân tích kỹ thuật (technical analysis) trong thị trường tài chính. Tôi dẫn seminar này, dựa trên kinh nghiệm phân tích . . . → Read More: Phân tích kỹ thuật (1)

Topology and stability of integrable Hamiltonian systems

There is a new survey paper by A. Bolsinov, A. Borisov and I. Mamaev about the topology and stability of integrable Hamiltonian systems in Russian Mathematical Surveys (Vol. 65, 2010, No. 2, pp  259–318). A copy of it can be downloaded here. In this paper, the authors discuss the topology of (finite dimensional) integrable . . . → Read More: Topology and stability of integrable Hamiltonian systems

Notes on INS (4): Useful inequalities & spaces

Last updated: 25/Oct/2010

There are two many inequalities used in the theory of Navier-Stokes equations. I’ll have to keep track of them. So this is the place where I put the inequalities.

Gagliardo-Nirenberg-Sobolev  (GNS). Assume . Then there exists a positive constant , depending only on and , such that for all (space of . . . → Read More: Notes on INS (4): Useful inequalities & spaces

Amusing dog quotes

Once a dog, always a dog

- I love a dog. He does nothing for political reasons. ~Will Rogers

- The more one gets to know of men, the more one values dogs. ~Alphonse Toussenel

- Women and cats will do as they please and men and dogs should relax and get used . . . → Read More: Amusing dog quotes

Notes on INS (3): Koch-Tataru theorem (small initial data)

Last updated: 16/Oct/2010

Existence of smooth global solutions for small initial conditions

The strongest (and in a sense optimal ?) result in this direction is due to Koch and Tataru (2001).

Reference: H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equation, Adv. Math. 157 (2001), No. 1, 22–35.

See also: P. Germain, N. Pavlovic, . . . → Read More: Notes on INS (3): Koch-Tataru theorem (small initial data)

A picture of Poisson 2010

Winter 2010, IMPA, Rio de Janeiro

Click on the image to see it in large size (and try to guess who is who )

Notes on INS (2): non-existence of self-similar solutions

Non-existence of self-similar solutions

Last updated: 13/Oct/2010

See the first part here: Notes on INS (1)

In this part, we will look at the proof of the non-existence of self-similar singular solutions to the INS equation.

The main reference is: Neças, Ruczicka and Sverak (Acta Math., Vol 196, 1996)

We will first follow the . . . → Read More: Notes on INS (2): non-existence of self-similar solutions

Notes on INS (1): in the beginning …

Last updated: 30/Oct/2010

INS = incompressible Navier-Stokes equation:

, where and

These are the notes that I’ll write about this problem, in order to learn about it and keep track of it.

Unless otherwise indicated explicitly, the domain will be , no external field, the initial condition (velocity field) will be smooth and . . . → Read More: Notes on INS (1): in the beginning …

Google created cars which can drive themselves in heavy traffic

News from The New York Times:

http://www.nytimes.com/2010/10/10/science/10google.html

Automatic driving must use “geometric control theory”, a branch of mathematics.  Now Google is invading it too. This “Google Monopoly” is becoming really “dangerous” !

Vai trò của hài hước trong giảng dạy

Có một định lý trong giáo dục học là:

Tâm trạng của học sinh khi học ảnh hưởng rất lớn đến khả năng tiếp thu và kết quả học tập.

Trạng thái tâm lý thuận lợi cho việc học là: thoải mái, vui vẻ, phấn khích, tập trung.Học mà vui sướng, tập trung và . . . → Read More: Vai trò của hài hước trong giảng dạy

Copula trong xác suất và tài chính (3)

Phần tiếp nối bài  Copula trong xác suất và tài chính (2). Phần này bàn về một số họ copulas

Marshall-Olkin copulas

Họ copula này được Marshall & Olkin nghiên cứu từ quãng năm 1967 (Ref: Marshall, A., and I. Olkin (1967): “A multivariate exponential distribution,” Journal of the American Statistical Association, 62, 30–44), . . . → Read More: Copula trong xác suất và tài chính (3)