Asymptotic behavior of solutions on modifications of three-dimensional Navier–Stokes equations with unbounded delays

Abstract

We consider modifications of 3D Navier–Stokes equations involving unbounded delays in $\mathrm{\Omega }\subset {ℝ}^3$. First, we show the existence and uniqueness of weak solutions by the Galerkin approximations method and energy method. Second, we prove the existence and uniqueness of stationary solutions by the Brouwer fixed point theorem. Finally, we study the stability of stationary solutions by the classical directly approach and construction of Lyapunov function. We also give a sufficient condition for the polynomial stability and polynomial decay of the stationary solutions in some special cases of unbounded variable delays.