I want to ask what advantage of using vorticity equations in fluid dynamics. Does it help to find large curls? Does it have singularities connected to presence of curls?
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The vorticity equation for the Euler equation in 3D is, with $\omega=\text{curl } v$, $$ \dot\omega + (v\cdot\nabla)\omega-(\omega\cdot\nabla)v=0, $$ so that if $v$ is two-dimensional, i.e. $ v=\begin{pmatrix}v_1(x_1, x_2)\\ v_2(x_1, x_2)\\ 0\end{pmatrix} $ you get that $$ \text{curl } v=\begin{pmatrix}0\\ 0\\ \partial_1v_2-\partial_2 v_1\end{pmatrix} \quad \text{so that}\quad \omega\cdot \nabla= (\partial_1v_2-\partial_2 v_1)\partial _3 \quad \text{and }\quad(\omega\cdot \nabla) v=0, $$ yielding $\dot\omega + (v\cdot\nabla)\omega=0$ which is simply a transport equation from which you get (under mild assumptions on $v$) $$ \Vert{\omega (t, \cdot)}\Vert_{L^\infty}=\Vert{\omega (t=0, \cdot)}\Vert_{L^\infty}, $$ yielding existence and uniqueness results for the 2D Euler equation. This argument breaks down in 3D, since, even for the Navier-Stokes equation, you get the vorticity equation $$ \dot\omega + (v\cdot\nabla)\omega+\nu\ \text{curl}^2 \omega =(\omega\cdot\nabla)v. $$ On the other hand it remains an elegant way of getting rid of the pressure since you can easily verify that $ \text{div}\bigl((v\cdot\nabla)\omega-(\omega\cdot\nabla)v\bigr)=0 $ (true since the commutator of vector fields with null divergence has also null divergence). By the way, you also get rid of the Leray-Hopf projection by writing the vorticity equation.
