I am trying to solve the vorticity equation, Eq. 9 in this paper:
\begin{aligned} \partial_{t} w(x, t)+u(x, t) \cdot \nabla w(x, t) &=\nu \Delta w(x, t)+f(x), & & x \in(0,1)^{2}, t \in(0, T] \\ \nabla \cdot u(x, t) &=0, & & x \in(0,1)^{2}, t \in[0, T] \\ w(x, 0) &=w_{0}(x), & & x \in(0,1)^{2} \end{aligned}
I have tried to play around with various methods, but apparently none of them is working. I think the difficulty is related to the fact that the equation contains a third order spatial derivative, and I am not sure how to proceed with this kind of problems. Does anyone know a solution to this kind of issue (or can suggest any useful reference)?