On the asymptotic behaviour and stochastic stabilization of second grade fluids

ABSTRACT

In this article, we study stability properties for two-dimensional non-Newtonian fluids. More precisely, we consider stochastic perturbations of the second grade fluid equations, with Navier slip boundary condition, and analyse the asymptotic behaviour of the solutions as $t\to +\mathrm{\infty }.$ We prove that the strong solutions (in the probability sense) of the stochastic evolutionary equation converge exponentially to the stationary solution in the mean square and almost surely. In addition, we study the stabilization of the deterministic model by introducing a suitable stochastic noise.

KEYWORDS:

• Second grade fluids
• stability
• stabilization
• stochastic

AMS SUBJECT CLASSIFICATIONS:

• 76A05
• 93D20
• 76F55
• 76M35

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matemática e Aplicações).