Asymptotic expansion conjecture for stationary Navier–Stokes flows with non-zero net force

Let $\mathbf{u}^{*}$ be boundary data for the body problem, or let $\boldsymbol{f}$ be a source term, and suppose that the corresponding net force $\boldsymbol{F}$ is non-zero. Write $\boldsymbol{u}=(u,v)$, $\omega=\boldsymbol{\nabla}\boldsymbol{\wedge}\boldsymbol{u}$, and $r=|\boldsymbol{x}|$. For $i\in\{0,1\}$, let $u_i$, $v_i$, $p_i$, and $\omega_i$ denote the terms in the asymptotic expansion, with $a$ the parameter linked to $F$ by the stated relation.

Asymptotic expansion conjecture. For a large class of boundary conditions $\mathbf{u}^{*}$, respectively source terms $\boldsymbol{f}$, there exists a solution of the body problem, respectively the source-force problem, satisfying

$$u=u_0+u_1+O(r^{-1}),\qquad p=p_0+p_1+O(r^{-2}),$$ $$v=v_0+v_1+O(r^{-4/3}),\qquad \omega=\omega_0+\omega_1+O(r^{-5/3}).$$

Moreover,

$$u_i=O(r^{-(1+i)/3}),\qquad v_i=O(r^{-(2+i)/3}),\qquad p_i=O(r^{-(4+i)/3}),\qquad \omega_i=O(r^{-(3+i)/3})$$

for $i\in\{0,1\}$, and these functions depend only on the net force $\boldsymbol{F}$. In coordinates with $\boldsymbol{F}=(F,0)$ and $F>0$, the asymptotes are given by the order-zero formulas with $b_0=0$ and the order-one formulas with $b_1=0$; the parameter $a$ is linked to $F$ through the stated relation.

This conjecture proposes universal far-field asymptotics determined solely by the net force, for a large class of boundary conditions or source terms. The existence of such solutions and the asserted expansion remain open in the non-zero-net-force stationary two-dimensional exterior-domain setting.