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

Abstract

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 $\omega (x)=o(|x{|}^{-(1/q+1/q^2)})$ and $\nabla v(x)=o(|x{|}^{-(1/q+1/q^2)}\log |x|)$ as $|x|\to \infty$, where $\omega \equiv \mathrm{rot}v$. As an application, we prove the Liouville type theorem under the assumption that $\omega \in L^q({\mathbb{R}}^2)$. 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 $\omega \in L^q({\mathbb{R}}^2\times I)$.

14 pages. The result of the asymptotic behavior of the derivative of velocity is improved