Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

We consider the stationary and non-stationary Navier-Stokes equations in whole plane $\mathbb{R}^2$ exterior domain outside of large circle. The solution $v$ is handled class with $\nabla v \in L^q$ for $q \ge 2$. Since we deal case 2$, our larger sense spatial decay at infinity than that finite Dirichlet integral, i.e., $q=2$ where a number results such as asymptotic behavior solutions have been observed. For problem shall show $\omega(x)= o(|x|^{-(1/q + 1/q^2)})$ v(x) = o(|x|^{-(1/q+1/q^2)} \log |x|)$ $|x| \to \infty$, $\omega \equiv {\rm rot\,} v$. As an application, prove Liouville type theorem under assumption L^q(\mathbb{R}^2)$. problem, generalized $L^q$-energy identity clarified. also apply it to uniqueness Cauchy ancient L^q(\mathbb{R}^2 \times I)$.