Posted by: cnpde | September 21, 2012


Recently I begin to touch dynamical systems. My motivation is for PDE, but let me delay it for a while. Here I just want to talk a little about my feeling on DS and PDE.

After these days learning, I have a strong feeling that PDE is a mass (I do not want to be rude here, so I did not use the word ‘shit’)! I say so because I found dynamical system is such a beautiful subject. It can describe the evolution system in such a detailed way, which can never be reached for PDE. In dynamical systems, the properties we care includes repeatability and stability. The phenomenon that just happen once and will not be repeated is not the centre of research. Judging from the history of DS, we can understand this easily. The dynamic systems used to be developed for understanding the motion of planet. Most planets have periodic orbits. They will ‘visit’ earth in some definite frequency. That’s why people care about the phenomenon that can be repeated.

After people realized the earth is moving around the sun, a concern is spread among some educated persons, that is, will the earth falling to the sun? The stability of dynamic system comes from this question. In the theory of hyperbolic dynamical systems, the manifold can be divided into two or three parts: stable part, unstable part, (sometimes we may have) central part. This properties can be preserved under a map, called Anosov map. If this map is a diffeomorphism, we can get structural stability, which is helpful to classify dynamic systems. More detailed analysis may reveal that many dynamical phenomena are only related with topological properties, not with differential properties. So much energy has been put into study of topological dynamical systems. Mathematicians have constructed a great mansion to describe the phenomenon in a very detailed way.

However, when we get back to PDE, we may find the situation is really dispointing. In PDE, at this moment, people is still struggling on the well-posedness. This is good, because mathematicians are usually very careful for the objects that may not exist. Except that, what we know for a system is rather limited. We may get some assymptotic properties, and also local wellposedness, but, what is the middle part? We may see it clearly through following example. One of the million dollar problem is global wellposedness of Navier-Stokes equation, which is used to describe the motion of fluid. Currently, we have some results for local wellposedness, and also some assymptotic estimate. But we know almost nothing about the situation of middle part (This may be improper, because some dynamical results have been obtained, such as bifercation, etc, but still very limited). If we think in this way, then we may find the turbulence is just something happen in the middle time period. But we cannot describe it by NS equation (Here I do not consider whether this model is good or not). So PDE theory is far from being perfect, especially for nonlinear PDE. It has been called ‘dirty math’, because we have no general mathods to deal with nonlinear PDEs.

So much about the frustrating side. Above discussion also bring us good news. There are still many open problems in PDE, and we need to introduce new tools, new ideas into PDE research to help us understand the solution better!


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