Bifurcations of Solutions of the 2-Dimensional Navier–Stokes System
For the 2-dimensional Navier–Stokes System written for the stream functions we construct a set of initial data for which initial critical points bifurcate into three critical points. This can be interpreted as the birth of new viscous vortices from a single one. In another class of solutions vortices merge, i.e. the number of critical points decrease.
KeywordsInitial Data Periodic Orbit Extremal Point Stream Function Degenerate Case
The authors thank V. Yakhot for useful remarks and discussions. The first author is supported in part by a start-up fund from University of British Columbia. The financial support from NSF, grant DMS 0908032, given to the first author and grant DMS060096, given to the second author are highly appreciated.
- 1.V.I. Arnold, Lectures on bifurcations and versal families. A series of articles on the theory of singularities of smooth mappings. Uspehi Mat. Nauk 27 5(167), 119–184 (1972)Google Scholar
- 2.E. Dinaburg, D. Li, Ya.G. Sinai, Navier–Stokes system on the flat cylinder and unit square with slip boundary conditions. Commun. Contemp. Math. 12(2), 325–349 (2010)Google Scholar
- 3.C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989)Google Scholar
- 4.J.C. Mattingly, Ya.G. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier–Stokes equations. Commun. Contemp. Math. 1(4), 497–516 (1999)Google Scholar