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In this paper, we are concerned with bilinear estimates related to thetwo-dimensional stationary Navier--Stokes equation. By establishing concretecounter examples, we prove the bilinear estimates fail for almost all scalingcritical Besov spaces. Our result may be closely related to an open problemwhether the two-dimensional stationary Navier--Stokes equation on the wholeplane $\mathbb{R}^2$ is well-posed or ill-posed in scaling critical Besovspaces.

Counter examples for bilinear estimates related to the two-dimensional stationary Navier--Stokes equation