On time periodic solutions, asymptotic stability and bifurcations of Navier-Stokes equations

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• 作者：Chun-Hsiung Hsia ; Chang-Yeol Jung ; Thien Binh Nguyen…
• 关键词：Mathematics Subject Classification35Q30 ; 76D05 ; 37Gxx ; 65C20 ; 37N10
• 刊名：Numerische Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：135
• 期：2
• 页码：607-638
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
• 出版者：Springer Berlin Heidelberg
• ISSN：0945-3245
• 卷排序：135

文摘

In this article, we investigate the time periodic solutions for two-dimensional Navier-Stokes equations with nontrivial time periodic force terms. Under the time periodic assumption of the force term, the existence of time periodic solutions for two-dimensional Navier-Stokes equations has received extensive attention from many authors. With the smallness assumption of the time periodic force, we show that there exists only one time periodic solution and this time periodic solution is globally asymptotically stable in the \(H^1\) sense. Without smallness assumption of the force term, there is no stability analysis theory addressed. It is expected that when the amplitude of the force term is increasing, the time periodic solution is no longer asymptotically stable. In the last part of the article, we use numerical experiments to study the bifurcation of the time periodic solutions when the amplitude of the force is increasing. Extrapolating to the heating of the earth by the sun, the bifurcation diagram hints that when the earth receives a relatively small amount of solar energy regularly, the time periodic fluid patterns are asymptotically stable; while/when the earth receives too much solar energy even though in a time periodic way, the time periodic pattern of the fluid motions will lose its stability.