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