
By NTZung, on October 16th, 2010%
A list of other references and results (in no particular order)
1) A. Vasseur, Regularity criterion for 3d navierstokes equations in terms of the direction of the velocity, Applications of Mathematics,Volume 54 (2009), Number 1, 4752.
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)
By NTZung, on October 16th, 2010%
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 2dimensional 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 NavierStokes
By NTZung, on October 16th, 2010%
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
By NTZung, on October 16th, 2010%
Last updated: 28/Oct/2010
This part:
– L5/3 regularity of pressure (Sohrvon Wahl theorem)
– L10/3 regularity of velocity field.
References:
– Sohr & von Wahl
– Robinson’s lecture notes
– … ?
(to be added)
By NTZung, on October 13th, 2010%
Last updated: 25/Oct/2010
There are two many inequalities used in the theory of NavierStokes equations. I’ll have to keep track of them. So this is the place where I put the inequalities.
GagliardoNirenbergSobolev (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
By NTZung, on October 12th, 2010%
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, Wellposedness for the NavierStokes equation, Adv. Math. 157 (2001), No. 1, 22–35.
See also: P. Germain, N. Pavlovic, . . . → Read More: Notes on INS (3): KochTataru theorem (small initial data)
By NTZung, on October 11th, 2010%
Nonexistence of selfsimilar 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 nonexistence of selfsimilar 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): nonexistence of selfsimilar solutions
By NTZung, on October 10th, 2010%
Last updated: 30/Oct/2010
INS = incompressible NavierStokes 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 …
By NTZung, on September 3rd, 2010%
Complex means multiagent
Often no leader, only local interactions, but exhibits large scale structures (selforganization, emergence)
Continue reading Modeling complex systems by macroscopic models (lecture by Pierre Degond, 03/09/2010)
By NTZung, on September 2nd, 2010%
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
By NTZung, on September 2nd, 2010%
This model was introduced by Schelling:
T. Schelling, Dynamic models of segregation, J. Math. Sociol. 1 (1971), 143186
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)
By NTZung, on September 1st, 2010%
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


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