Can someone explain to me why the vorticity flux is conserved for a solution to navier stokes equation in 2D ? Ie why $\int_{\mathbb{R}^2} w(x,t) dx =cst$ if $w$ satisfy the vorticity equation for navier stokes system: $\frac{d}{dt}w + (v\cdot \nabla) w = (w\cdot \nabla)v + \Delta w$ with $v$ divergence free.
If one integrate the equation then he gets $\frac{d}{dt}\int_{\mathbb{R}^2} w dx = \int_{\mathbb{R}^2} \Delta w dx$ the right hand side should not be necessarily zero am I right ?