Localization of solutions for planar Navier-Stokes equations

We study two models of planar stationary flows of an incompressible homogeneous fluid in a semi-infinite strip $\Omega=(0,\infty)\times(0,L)$, $L\gt 0$. The first model corresponds to a non-standard Stokes system \[\begin{gather}
& -\nu \Delta \boldsymbol{u}= \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u})-\nabla p,\quad \operatorname{div} \boldsymbol{u}=0\quad\text{in } \Omega, \label{eq:1:604} \\
& \boldsymbol{u}=\boldsymbol{u}_\ast \text{ for } x=0,\quad u=0 \text{ for } y=0, L,\label{eq:2:604} \\
& \boldsymbol{u}\to 0  \text{ as }   |\boldsymbol{x}|\to \infty,  \label{eq:3:604}\end{gather}\] where $\boldsymbol{x}=(x,y)\in\mathbb{R}^2$, $\boldsymbol{u}(\boldsymbol{x})=(u(\boldsymbol{x}),v( \boldsymbol{x}))$ is the velocity vector field, $p=p(\boldsymbol{x})$ is the hydrostatic pressure divided by the constant density of the fluid and $\nu$ is the kinematics viscosity coefficient. The body forces are given in a feedback dissipative form, $f:\Omega\times \mathbb{R}^2\to \mathbb{R}^2$, $\boldsymbol{f}=(f_1,f_2)$, such that for all $\boldsymbol{u}\in \mathbb{R}^2$ and almost all $x\in\Omega$ \[\begin{equation}-f_1(\boldsymbol{x},\boldsymbol{u})u\geq \delta  |u|^{1+\sigma}\quad    \text{for some } \delta\gt 0, \sigma \in (0,1) \label{eq:4:604}\end{equation}\] and \[\begin{equation}\operatorname{supp}f_2\cap\Omega^{x_g}\times\mathbb{R}^2=\emptyset \text{ for some } x_g\in(0,\infty),\quad   \Omega^{x_g}=(x_g,\infty)\times(0,L). \label{eq:5:604}\end{equation}\]

Because this kind of forces field is new in the Fluid Mechanics setting, we start by proving the existence of, at least, one weak solution for this problem. Then, we prove an uniqueness result under a non-increasing condition on the forces field. Finally, we prove the weak solutions of ($\ref{eq:1:604}$-$\ref{eq:3:604}$) with $\boldsymbol{f}$ satisfying ($\ref{eq:4:604}$-$\ref{eq:5:604}$) have compact support in $\Omega$, which means, from the physical point of view, this fluid can be stopped at a finite distance from the strip entrance.

Then, we extend these results for the second model which will be studied here, a non-standard Navier-Stokes system \[\begin{gather*}    -\nu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}= \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u})-\nabla p,\quad   \operatorname{div} \boldsymbol{u}=0    \quad   \text{in }   \Omega, \\ \boldsymbol{u}=\boldsymbol{u}_* \text{ for } x=0, \quad  \boldsymbol{u}=0 \text{ for }y=0, L, \\ \boldsymbol{u}\to 0  \text{ as }  |x|\to\infty,\end{gather*}\] where the forces field also satisfies ($\ref{eq:4:604}$-$\ref{eq:5:604}$).

If there is enough time, we will talk also about the same kind of localization effects for a stationary non-standard Boussinesq system and for the evolutionary systems.