You can recognize a pioneer by the arrows in his back.
by Beverly Rubik

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

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)

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

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)

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 …

Modeling complex systems by macroscopic models (lecture by Pierre Degond, 03/09/2010)

Complex means multi-agent

Often no leader, only local interactions, but exhibits large scale structures (self-organization, emergence)

Continue reading Modeling complex systems by macroscopic models (lecture by Pierre Degond, 03/09/2010)

Math model of Romeo and Juliet

An  amusing illustration of qualitative dynamical systems by J.C. Sprott based on a simple model of love between Romeo and Juliet:

http://sprott.physics.wisc.edu/lectures/love&hap/sld001.htm

Even though the model is very very simple, there’s already a great number of possibilities (different dynamics) !

Good teaching material for a first course on differential equations and mathematical modeling.

There . . . → Read More: Math model of Romeo and Juliet

Schelling’s segregation model (quick notes)

This model was introduced by Schelling:

T. Schelling, Dynamic models of segregation, J. Math. Sociol. 1 (1971), 143-186

T. Schelling, Micromotives and macrobehavior (1978)

Interactions among agents (micromotives) may lead to strong segragation effects (macrobehavior). Implications for social and economic policies aiming at fighting urban segregation (ghettos).

The model is as follows: agents of . . . → Read More: Schelling’s segregation model (quick notes)

Losers theorem & the winner’s curse

There are two very simple mathematical theorems, which have lots of practical implications.

The first one is The Losers Theorem: in any competition, most competitors are losers. The strong form of the theorem (for strong competitions) reads: second place is the first loser.

Talking about fame: Everyone knows that there is a guy called . . . → Read More: Losers theorem & the winner’s curse