What is kth vortex formula?

Question

I want to study the kth vorticity equation. The NS equation is provided as

\begin{align}\label{eq1} &\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \right)\textbf{u} = \nu \bigtriangleup \textbf{u} -\dfrac{1}{\rho} \textbf{Grad}(p) \end{align}

which implies

\begin{align}\label{eq2} &\dfrac{\partial }{\partial t} \textbf{curl}^k(\textbf{u}) + \textbf{curl}^k\biggl(\left(\textbf{u}\cdot \nabla \right)\textbf{u}\biggr) = \nu~ \bigtriangleup \textbf{curl}^k( \textbf{u}) \tag{1} \end{align}

with $$\textbf{curl}^k(\textbf{u}(x,t)) = \textbf{curl}(\textbf{curl}\cdots \cdots \textbf{curl})(\textbf{u}(x,t))$$ where we apply the curl $k$ times in succession. I want to compute $\textbf{curl}^k\biggl(\left(\textbf{u}\cdot \nabla \right)\textbf{u}\biggr)$ for each $k$. We know for $k=1$ we have $$\dfrac{\partial }{\partial t} \textbf{v} + \biggl((\textbf{u}\cdot \nabla )\textbf{v} -(\textbf{v}\cdot \nabla )\textbf{u} \biggr) = \nu~\bigtriangleup \textbf{v} , ~~~~~ \textbf{v}= \textbf{curl}(\textbf{u}).$$ Is it true in general $$\dfrac{\partial }{\partial t} \textbf{v}^k + \biggl((\textbf{u}\cdot \nabla )\textbf{v}^k-(\textbf{v}^k\cdot \nabla )\textbf{u} \biggr) = \nu~\bigtriangleup \textbf{v}^k, ~~~ \textbf{v}^k:= \textbf{curl}^k(\textbf{u}(x,t))$$

Answer 1

Use that the velocity field is incompressible, $\nabla\cdot u=0$, to rewrite $$(u\cdot\nabla)u_j=\sum_{i} \nabla_i (u_iu_j).$$ You seek the curl of the curl of this expression, use that $$[\operatorname{curl}\operatorname{curl}f]_j=\sum_{i}\bigl(\nabla_j\nabla_i f_i-\nabla_i\nabla_if_j\bigr).$$ Hence $$[\operatorname{curl}\operatorname{curl} (u\cdot\nabla u)]_j=\sum_{i}\sum_k\nabla_i\nabla_k\bigl(\nabla_j(u_ku_i)-\nabla_i (u_ku_j)\bigr).$$

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To check that the conjectured formula in the OP does not hold, consider the divergence-free velocity field $u=(x+y,x-y,xy)$. On the one hand, $\operatorname{curl}\operatorname{curl} u=0$, so the formula in the OP gives 0, but $\operatorname{curl}\operatorname{curl} (u\cdot\nabla)u=(0,0,-4)$ is nonzero.