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# The exponential behavior of Navier-Stokes equations with time delay external force

• In this paper we discuss the existence and the exponential behaviour of the solutions to a 2D-Navier-Stokes equation with time delay external force $f(t-\tau(t),u(t-\tau (t))),$ where $f(t,u)$ is a locally Lipschitz function in $u$ and $|f(t,u)|^2\leq a|u|^k+b_f,$ $a>0,b_f\geq 0,k\geq 2.$ $\tau (t)$ is a differentiable function with $0\leq \tau (t)\leq r, r>0,\frac{d}{dt}\tau (t)\leq M<1,$ $M$ a constant. We show the relations between the kinematic viscosity $\nu ,$ time delay $r>0$ and $\lambda_1, a, b_{f}, k, M$ play an important role. Furthermore, we consider the exponential behaviour of the strong solutions to a 3D-Navier-Stokes equation with time delay external force $f(t-\tau(t),u(t-\tau (t))),$ where $f(t,u)$ is a locally Lipschitz function in $u$ and $|f(t,u)|^2\leq a|u|^2+b_f,$ $a>0,b_f\geq 0.$ We extend Corollary 64.5[11]. Furthermore we discuss the existence of a periodic solution.
Mathematics Subject Classification: 35Q35, 35Q30, 35B10.

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