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 R² and in the exterior domain outside of the large circle. The solution v is handled in the class with ∇v ∈ L^q for q ≥ 2. Since we deal with the case q ≥ 2, our class is larger in the sense of spatial decay at infinity than that of the finite Dirichlet integral, that is, 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^/q2)) and ∇v(x) = o(|x|⁻⁽1/q+1^/q2)log|x|) as |x| → ∞, where ω ≡ rotv. As an application, we prove the Liouville-type theorem under the assumption that ω ∈ L^q(R²). 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 ∈ L^q(R² × I).