Recently, I've been working with a colleage on the numerical solution of some problems in fluid mechanics via the Van-Kan algorithm.
In the case we're studying these days, we've come across the following problem:
$$\left\lbrace \begin{array}{l} \partial_x G_1+ \partial_y G_2 = 0, \quad \text{in } \Omega\\ G_1(0,y) = y(2-y), \quad y \in [0,2], \\ G_1(x,y) = 0, \quad \text{in } \partial \Omega \setminus \{x = 0\}, \\ G_2(x,y) = 0, \quad \text{in } \partial \Omega, \end{array} \right.$$ where $G = (G_1,G_2) \colon \mathbb{R}^2 \to \mathbb{R}^2$ and $\Omega = [0,10] \times [0,2]$.
We are stuck in deciding whether if the upper problem has solution or not. We've notice that, if the solution exists, it must not be compatible with the Divergence/Green theorem (as the given boundary conditions do not integrate to $0$ on $\partial \Omega$). Any advice on the (non-)existence of solutions?
Thank you in advance.
There is no solution, as the divergence theorem can be extended to Sobolev spaces:
Michel Willem - Functional Analysis (Fundamentals and Applications) - Birkhäuser Basel (2013). Theorem 6.3.4.