# Boundary flux maximizing drift (velocity) vector fields for 2D heat equation

Looking for literature / known results on the following class of problems:

Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar field $T=T(x,t)$ with no slip and steady Dirichlet conditions: $u(\partial\Omega,t)=0\:\forall t,\: T(\partial\Omega,t)=T_0$

The scalar field evolves as:

$\frac{\partial T}{\partial t}+u.\nabla{T}=k\nabla^2{T}$

Now consider the functional:

$F(u)=\int_0^{t_1}\int_{\partial\Omega} \frac{\partial T}{\partial \hat n}ds dt$, where $\hat n$ is the outward normal on the boundary. This functional measures the total flux out of the domain in some finite time $t_1$. The aim is to maximize this functional within a class of permissible $u$ with some bounded norm (energy), say for example divergence free $u\in L^2(\Omega)$ and $||u||_{L^2}=1$.

My question is that whether this problem has been looked at in the PDE literature before, and if yes, what are the known results. For example, what is the relation between the shape of $\Omega$ and the optimal $u$.

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