Background: I am aware that in for example physics tensor fields can be used to describe things like properties of materia. Like heat conduction in macroscopic media (imagine a thermos, heat can flow along the walls but much less across walls) or electrical conduction (electrons can flow easily along a copper wire but not at all so easily from the wire to the surrounding air).
In these examples Electrical currents are vectors and temperatures are scalars. So we have examples of both 1-tensors and 0-tensors which in both cases a order 2-tensor field can describe how they will behave.
I can't find the references so I will derive energy minimization approach from memory. Wikipedia states heat equation as :
$$\frac{\partial u}{\partial t} - \alpha \nabla^2 u = 0$$ But in reality we need to look closer, it is actually a divergence of the gradient : $$\frac{\partial u}{\partial t} - \alpha \nabla\cdot(\nabla u) = 0$$ And if $\alpha \notin \mathbb R$, but instead ${\bf A \in \mathbb R}^{2\times 2}$: $$\frac{\partial u}{\partial t} - \nabla\cdot( {\bf A} \nabla u) = 0$$
${\bf A}$ is here the orientation (or maybe called anisotropy in physics?) tensor.
We can now make this an energy minimization problem by doing:
$$u_o = \min_{u}\left\{ \int_{\Omega} \left\|\frac{\partial u}{\partial t} - \nabla\cdot( {\bf A} \nabla u)\right\|d \Omega\right\}$$
Now to my question, do there exist examples of order 2 (or higher) tensor fields which are "steered" by tensor fields of order 2 in some similar sense?
For example a field of matrix operators which are steered by tensor field? Can we think of example where some linear operator describable by unique matrix in every point is "steered" by a 2-tensor?
In some kind of similar notation to the above:
$${\bf U_o} = \min_{\bf U}\left\{ \int_{\Omega} \left\|\frac{\partial \bf U}{\partial t} - ( {\bf A} {\bf U} {\bf A}^T)\right\|d \Omega\right\}$$
Since I don't have any example this is just some guess of functional, I don't know how it should be interpreted or anything.