Nonlinear stability and asymptotic behavior of the viscoelastic Rayleigh–Bénard problem with some classes of large perturbations

This paper investigates the global (in time) stability and asymptotic behavior of solutions near equilibrium to the Rayleigh–Bénard problem of incompressible viscoelastic fluids within a three-dimensional horizontal layer domain. We adapt our analysis to Lagrangian coordinates and demonstrate the global stability of solutions for some classes of large initial perturbations, whenever the elastic coefficient is sufficiently large relative to the large initial perturbations. Additionally, the solution exhibits an exponential time decay towards its equilibrium. Furthermore, we show that the nonlinear problem, which exhibits algebraic convergence rates with respect to the elastic coefficient, converges to the corresponding linearized problem as the elastic coefficient approaches infinity. Finally, we extend the aforementioned results to the case without heat conductivity. Our results provide insight into the stabilizing effect of elasticity in inhibiting thermal instability in viscoelastic fluids and the asymptotic behavior of such systems, especially in the context of large perturbations.