Define the associated vorticity field \( \mathbf {w}=(w_1,w_2,w_3)=curl \mathbf {v}=\nabla \times \mathbf {v} \) , which describes the rotation of the fluid. Then \( \mathbf {w} \)  satisfies the vorticity equations (see [1]}, [2]})
\({\left\lbrace \begin{array}{ll}\partial _t\mathbf {w}+(\mathbf {v}\cdot \nabla )\mathbf {w}=(\mathbf {w}\cdot \nabla )\mathbf {v},\ \ &D\times (0,T),\\\mathbf {w}(\cdot , 0)=\nabla \times \mathbf {v}_0(\cdot ), \ \ &D.\end{array}\right.}\) 

Зададим векторное поле завихренности \( \mathbf {w}=(w_1,w_2,w_3)=\text{curl} \mathbf {v}=\nabla \times \mathbf {v} \), описывающее вращение жидкости. Тогда \( \mathbf {w} \) удовлетворяет уравнениям завихренности (см. [1], [2]):

\({\left\lbrace \begin{array}{ll}\partial _t\mathbf {w}+(\mathbf {v}\cdot \nabla )\mathbf {w}=(\mathbf {w}\cdot \nabla )\mathbf {v},\ \ &D\times (0,T),\\\mathbf {w}(\cdot , 0)=\nabla \times \mathbf {v}_0(\cdot ), \ \ &D.\end{array}\right.}\)