Condition for a vector field be non-linear

If a vector field ##\vec{v}## is non-divergent, so the identity is satisfied: ##\vec{\nabla}\cdot\vec{v}=0##;



if is non-rotational: ##\vec{\nabla}\times\vec{v}=\vec{0}##;



but if is "non-linear"





so the laplacian of v is zero?



##\nabla^2 \vec{v} = \vec{0}##