We study the exterior problem for stationary Navier-Stokes equations in two dimensions describing a viscous incompressible fluid flowing past an obstacle. We consider the classical questions about the existence of solutions, their asymptotic properties, etc., as well as recent results obtained in our joint papers with K.~Pileckas and R.~Russo, --- in particular, the nontriviality of Leray solutions, boundedness and uniform convergence of arbitrary D-solutions (i.e., solutions with finite Dirichlet integral), etc.

In the last part of the talk we discuss the very recent result obtained in our joint paper with Xiao Ren (任潇, Fudan University). Namely, it is shown that, at small Reynolds numbers, the classical solutions constructed by Finn and Smith are unique in the class of D-solutions. No additional symmetry or decay assumptions are required. This result answers a long-standing open problem. In the proofs, we developed the ideas of the classical Ch.~Amick paper (Acta Math. 1988).

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