I am attempting to finite element model the/a solution to the Kuramoto-Sivashinsky equation
$$u_{t}+\nu u_{xxxx}+u_{xx}+uu_{x}=0 $$ with 1-periodic boundary condition $$u(\cdot,0)=u_{0} $$
using the software FreeFem++.
Using this paper, I derived the following weak form of the equation:
$$ (u_t,\phi)+\nu(u_{xx},\phi_{xx})-(u_x,\phi_x)+(uu_x,\phi) = 0 $$
with $\phi \in H_{0}^2(I)$ as defined in the paper linked above.
I am having trouble translating this weak form into FreeFEM++ code. I am unsure how to handle the higher order term $\nu(u_{xx},\phi_{xx})$ and the nonlinear term $(uu_x,\phi)$. Does anyone know how to represent this in FreeFEM? Will I need to discretize the weak form or is this good enough? Can the command int2d be used to solve this problem?