I am looking for results concering the following parabolic PDE $$u\cdot\nabla u + \Delta u = F(x),$$ where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or unbounded), preferably the torus ($\mathbb{T}^2$). I do not specify any boundary condition, as I'm interested in results for any of periodic, Dirichlet, Neumann bd.cond, or unbounded domain.

This equation can be written in scalar form as $$uu_x+vu_y+u_{xx}+u_{yy}=f\\uv_x+vv_y+v_{xx}+v_{yy}=g$$

This equation can be interpreted as time-independent 2D forced viscous Burgers' equation.

I am looking for any known qualitative theoretical results. I've been surprised as little is available in the literature. I searched the net a bit for any existence, uniqueness, multiplicity of solutions result for this equation , but ended up having a bunch of numerical papers, and none theoretical. It looks like this equation is often used as a test-case for numerical methods.

I am aware that the case $F(x)=0$ is simpler, as the equation reduces to a linear one by Hopf-Coles' transform, at least in some specific cases, and there are some results for that.

I am interested in the cases $\|F\|\neq 0$, and especially $\|F\|$ large.

I would also appreciate results for time-dependant version, i.e. $u_t=u\cdot\nabla u + \Delta u + F$.

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