Let $\Delta : \mathbb{R}^3\rightarrow \mathbb{R}^3$ 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 $$\dfrac{\partial w_i}{\partial t}+ |\textbf{u}|~|\textbf{w}| = \mu \Delta w_i,~~~~~ \textbf{w} = \operatorname{curl}\textbf{u}, ~~~~~1\le i\le 3$$
and $$\textbf{u}(x,0) = \textbf{u}_0(x)$$ at time $t=0$ along with the bounded energy condition
$$ \int_{\mathbb{R}^3}^{}{|\textbf{u}(x,t)|^2dx}<\epsilon,~~~~~~ \forall \epsilon > 0.$$ Here $w_i$ denotes the ith component of $\textbf{w}$. These equations are to be solved for $\textbf{w}$ and $\textbf{u}$. If a solution $\textbf{u}(\textbf{x},t)$ exist for all time $t>0$ is it unique and does it blow up?