Well-posedness and invariant measures for 2D stochastic Oldroyd model of order one with pure jumps

Abstract

In this paper, we consider the stochastic two-dimensional viscoelastic fluid flow equations driven by a multiplicative Lévy noise, arising from the Oldroyd model for the non-Newtonian fluid flows. The system carries the natural Lipschitz condition on balls and linear growth assumptions on the jump coefficient. We first prove the existence and uniqueness of a global strong solution by using stopping time technique and Banach fixed point theorem. Meanwhile, we overcome the difficulties posed by the jump and memory term, the latter makes some convergence findings more natural. By demonstrating the exponential stability of strong solutions, we establish the existence of a unique invariant measure for the stochastic system. Numerical results provide further support for these theoretical findings, the dissipative velocity of solution is more rapid and rigorous when compared with the stochastic Navier-Stokes equation.