The weak formulation of the following Stokes-like problem without Dirichlet boundary conditions is well posed?
Let $\Omega\subset\mathbb{R}^2$ a bounded domain.
The weak problem is: Find $u\in H^1(\Omega)^2$ and $p\in L^2(\Omega)$ such that
$$\left\{\begin{array}{l}\displaystyle\int_\Omega\nabla u:\nabla\,dx+\int_\Omega u\cdot v\,dx-\int_\Omega p\,div(v)=\int_\Omega f\cdot v\,dx+\int_{\partial\Omega}g\cdot v\,dx\quad\textrm{ for all }v\in H^1(\Omega)^2 \\ \displaystyle-\int_\Omega q\,div(u)=0\quad\textrm{ for all }q\in L^2(\Omega) \end{array}\right.$$
The strong problem is:
$$\left\{\begin{array}{rl}-\Delta u+u+\nabla p =f&\textrm{ in }\Omega \\ div(u)=0&\textrm{ in }\Omega \\ (\nabla u-p\,I)n=g&\textrm{ on }\partial\Omega \end{array}\right.$$
Note that this problem don't has Dirichlet boundary condition (don't has conditions like $u=g_D$ on $\partial\Omega$) and only has Neumann boundary condition (the condition $(\nabla u-p\,I)n=g$ on $\partial\Omega$)