Solutions of the stationary Navier-Stokes equations are studied in the entire plane with a force field having a compact support and with a given constant limit at infinity. Previously, the existence of solutions to this problem was known only under special assumptions about symmetry and smallness. In this paper, we solve the main difficulties in applying Jean Leray's "exhaustive domains" method and, as a consequence, we prove the existence of D-solutions in the entire plane for an arbitrary force with a compact support. The fulfillment of the limit conditions at infinity can be verified under two different scenarios:
(I) the terminal speed is sufficiently large in relation to the external force,
(II) both the total force integral and the limiting velocity vanish.
Thus, this method gives a large class of new solutions with given limiting velocities. In addition, the uniqueness of D-solutions of this problem is established under the condition that the force is small in comparison with the limiting velocity. The main tools here are two new general estimates for arbitrary solutions of the Navier-Stokes system, which have a fairly simple form. They control the difference in the average speeds over two concentric circles through the Dirichlet integral in the ring between them.
The report is based on a recent joint article with Julien Guillod (Sorbonne University, Paris) and Xiao Ren (Fudan University, Shanghai). See https://arxiv.org/abs/2111.11042.
