Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations

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• 作者：Jacek Cyranka^a ; ^b ; ^{jacek.cyranka@ii.uj.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Piotr Zgliczyński^a ; ^{piotr.zgliczynski@ii.uj.edu.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B40 ; 35Q30 ; 35B41
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：261
• 期：8
• 页码：4648-4708
• 全文大小：674 K

文摘

We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to the Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the viscous Burgers' equation, and the incompressible 2D Navier–Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier–Stokes equations we establish the existence of a locally attracting solution.