Algebraic bounds on the Rayleigh–Bénard attractor. (22nd January 2021)

• Record Type: Journal Article
• Authors: Cao, Yu; Jolly, Michael S; Titi, Edriss S; Whitehead, Jared P
• Is Part Of: Nonlinearity. Volume 34:Number 1(2021)
• Page Start: 509
• Page End: 531
• Publication Date: 2021-01-22
• Subjects: Rayleigh–Bénard convection -- global attractor -- synchronization; 35Q35 -- 76E06 -- 76F35 -- 34D06; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abb1c6
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.