Existence of wake-asymptotic solutions for two-dimensional Navier–Stokes flow

Consider the steady two-dimensional Navier–Stokes equations with boundary conditions $\boldsymbol{u}^{*}$ and source term $\boldsymbol{f}$, and let $\boldsymbol{F}$ denote the net force. Assume that $\boldsymbol{F}\neq\boldsymbol{0}$ and that the boundary conditions and source terms belong to a large class. Let $({\boldsymbol{U}\!}_{\boldsymbol{F}},P_{\boldsymbol{F}})$ be the solution constructed in the preceding proposition. Wake-asymptotic existence conjecture. There exists a solution with velocity at infinity $\boldsymbol{u}_{\infty}=\boldsymbol{0}$ satisfying

$$\boldsymbol{u}={\boldsymbol{U}\!}_{\boldsymbol{F}}+O(r^{-1}),\qquad p=P_{\boldsymbol{F}}+O(r^{-2}).$$

The constructed profile has a wake and decays like $|\boldsymbol{x}|^{-1/3}$ in the wake. The source presents this as a belief about a general asymptote for solutions with sufficiently small data; no proof or resolution is supplied.