Let $\textbf{A}_i\in M_2(\mathbb{R}^3\times [0,\infty))$ (the set of 2x2 matrices with real entries in $\mathbb{R}^3\times [0,\infty)$) for $1\le i\le 3$ be invertable $(\det(\textbf{A}_i)\neq 0$), $\bigtriangleup : \mathbb{R}^3\rightarrow \mathbb{R}^3$ denote the Laplacian operator, and $\mu\in \mathbb{R}$ with $\mu > 0$. Is there a vector field $$\textbf{u}:\mathbb{R}^3\times [0,\infty)\rightarrow \mathbb{R}^3$$ such that $$\dfrac{\partial w_i}{\partial t}+ |\textbf{u}|~|\textbf{w}|\det(\textbf{A}_i) = \mu \bigtriangleup w_i,~~~~~ \textbf{w} = \textbf{curl(u)},~~~~~1\le i\le 3$$
These equations are to be solved for a given initial velocity $$\textbf{u}(x,0) = \textbf{u}_0(x)$$ at time $t=0$ along with the bounded energy condition
$$ \int_{\mathbb{R}^3}^{}{|\textbf{u}(x,t)|^2dx}<\epsilon,~~~~~~ \forall \epsilon > 0.$$
The scalar $\mu$ is called the kinetic 'viscosity' (a quantity that measures the friction of a fluid). These equations are to be solved for the velocity field $\textbf{u}(x,t)$.
Here $w_i$ denoted the $i^{th}$ component of $\textbf{w}$. Now if there is a solution, is it unique and how does the solution depend on the matrices $\textbf{A}_i$, in other words how does changing $\textbf{A}_i$ alter the solution.
One thing I see interesting is the nonlinear term appears related to $$|\textbf{u}\times \textbf{w}| = |\textbf{u}||\textbf{w}|\sin(\alpha)$$ where $\alpha$ is the angle between $\text{u}$ and $\textbf{w}$