We consider the incompressible Navier-Stokes equations on the whole plane. In contrast to the initial value problem, the solvability of the 2D stationary Navier-Stokes equations has been open. In this talk, we solve this problem negatively in the scaling critical Besov spaces framework. Moreover, we apply our method to the time-periodic problem and show the nonexistence of time-periodic solutions for some small time-periodic external forces.