On the stability of the viscoelastic Bénard problem in some classes of large data

In this paper, we investigate the Bénard problem of the incompressible viscoelastic fluids heated from below in a three-dimensional periodic cell, and establish the global (-in-time) existence result of unique strong solutions whenever the elasticity coefficient is sufficiently large relative to both norms (of energy space of solutions) of the initial velocity and the initial perturbation temperature. Our new result mathematically verifies that the elasticity under the large elasticity coefficient can inhibit the thermal instability even if both the initial velocity and the initial perturbation temperature are large. Moreover, the solutions also enjoy the exponential decay-in-time. In addition, using the method of vorticity estimates, we further derive that the convergence rate of the nonlinear system towards a linearized pressureless problem, as either time or elasticity coefficient approaches infinity, is in the form of $c{\kappa }^{-1}$. Our converge rate is faster compared to the known rate $c{\kappa }^{-\frac{1}{2}}$ first found by Jiang–Jiang in [23].