| 内容 |
非線形性を持つ系はふつう一般的な解の表示を持たない．しかし系の漸近挙動 (たとえば時間無限大でのふるまい) には簡明で普遍的な構造の現れることが多い． このような漸近挙動を調べるためには，自己相似性に注目することが役に立つ． もっとも簡単な場合には相似則と相似変数は次元解析だけから求まるが， 一般に事情はより複雑であり，より一般的な相似則に着目することになる (この見方を Barenblatt に習って "Intermediate asymptotics" (IA) とよぶことにする[1])． まずこの考え方について具体例を通して説明する．
次に IA と関連の深い「くりこみ群の方法」(RG) を用いて，同種の問題に対して漸近挙動が 調べられることを説明する [2, 3]．RG でうまく解析できる現象がいくつか知られており (e.g. [4, 5])，数学としても話が進んでいる [6, 7, 8]．
Equations that describe nonlinear phenomena usually do not possess explicit formulae for their solutions. However, in the asymptotic behaviour there often appear surprisingly simple and universal laws. These universal laws have the property of self-similarity. In the simplest case, self-similarity and self-similar variables are obtained by a simple dimensional consideration. But this is only true in some restricted cases, and more general similarity must be considered. In this case, determination of self-similar variables leads to a nonlinear eigenvalue problem (we call this method "Intermediate asymptotics" (IA) following Barrenblatt [1]). We will explain this with an example.
Intimately related to IA, renormalization group (RG) method is also capable of deriving asymptotic laws [2, 3]. RG method has proved to be useful in analysing some physical phenomena (e.g. [4, 5]), and have been discussed in the mathematics literature [6, 7, 8]. We will explain the basic ideas of RG method with examples.
This seminar is a review of some papers from the reference below.
[1]: G. I. Barenblatt "Similarity, Self-Similarity, and Intermediate Asymptotics", Consultans Bureau, 1979.
[2]: N. Goldenfeld, O. Martin, Y. Oono "Intermediate Asymptotics and Renormalization Group Theory", Journal of Scientific Computing, Vol.4, No.4, 1989.
[3]: N. Goldenfeld, O. Martin, Y. Oono, F. Liu "Anomalous Dimensions and the Renormalization Group in a Nonlinear Diffusion Process", Phys. Rev. Lett., Vol.64, No.12, 1990.
[4]: 大田隆夫「繰り込み群の高分子系への応用」，高分子，33巻 6月号 1984年．
[5]: 原隆，古池達彦「爆発と凝集」(柳田英二 編，三村泰三 監修) 第四章「重力崩壊における臨界現象」， 東京大学出版会，2006年．
[6]: J. Bricmont, A. Kupiainen, G. Lin "Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations", Comm. Pure Appl. Math., Vol.XLVII, 1994.
[7]: H. Chiba "Approximation of Center Manifolds on the Renormalization Group Method", Journal of Mathematical Physics, Vol.49, 2008.
[8]: H. Chiba "C^1 Approximation of Vector Fields Based on the Renormalization Group Method", SIAM J. Appl. Dyn. Syst., Vol.7, 2008. |
