On the Forced Surface Quasi-Geostrophic Equation: Existence of Steady States and Sharp Relaxation Rates

2020/08/22 by Fazel Hadadifard, Atanas Stefanov, Atanas G. Stefanov · 2 citations
Chemistry · Mathematics · #Advanced Mathematical Physics Problems #Chemistry #Forcing (mathematics) #Function (biology) #Geometric Analysis and Curvature Flows #Geometry #Geostrophic wind #Infinity #Mathematical analysis #Mathematics #Mechanics #Navier-Stokes equation solutions #Physics #Relaxation (psychology) #Scaling #Steady state (chemistry) #Surface (topology) #math.AP #msc:35B40 #msc:35Q35 #msc:76B03 #msc:76D03 #msc:76D07

paper · pdf · doi:10.1007/s00021-021-00559-1

published in Journal of Mathematical Fluid Mechanics 23(1) (Birkhäuser)

arxiv created 2020/08/22 · openalex created_date 2020/09/01 · openalex publication_date 2021/02/01 · arxiv updated 2021/02/24 · openalex updated_date 2026/08/05

Abstract

We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all L¹∩ L^∞ localized initial data produces global solutions of the forced SQG, which converge to the steady states in L^p(\mathbf R²), 1<p≤ 2 as time goes to infinity. This establishes that the steady states serve as one point attracting set. Moreover, by employing the method of scaling variables, we compute the sharp relaxation rates, by requiring slightly more localized initial data.

Citations

• On the global well-posedness for Boussinesq system
• Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R 2
• Global regularity for the 2D Boussinesq equations with partial viscosity terms
• Long-time asymptotics of the Navier-Stokes and vorticity equations on R³