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 R-2 and in the exterior domain outside of the large circle. The solution v is handled in the class with del v is an element of 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 omega(x) = o(vertical bar x vertical bar(-(1/q+1/q2))) and del v(x) = o(vertical bar x vertical bar(-(1/q+1/q2)) log vertical bar x vertical bar) as vertical bar x vertical bar -> infinity, where omega = rot v. As an application, we prove the Liouville-type theorem under the assumption that omega is an element of L-q(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 is an element of L-q(R-2 x I).