Let $f = |u|^{p-1}u$ where $u(t,x):\mathbb{R}\times \mathbb{R}^d\to\mathbb{R}$ and $p>2.$ Define $G=f(Q_1+Q_2) - f(Q_1)-f(Q_2)$ where $Q_k(t,x)=Q(x-z_k(t))$ where $Q$ is a positive exponentially decaying function and $k=1,2.$ Then I want to find functions $g_1$ and $g_2$ such that, $$G = G^{\perp} + \sum_{k=1,2}g_k\cdot \nabla Q_k$$ where $\langle G^{\perp}, \nabla Q_k \rangle = 0$ for $k=1,2$ where $$\langle f,g\rangle = \int f(x)g(x){\rm{d}}x$$ and for vector valued functions $\vec{f} = (f_1,f_2)$ and $\vec{g} = (g_1,g_2)$ we have that, $$\langle \vec{f},\vec{g} \rangle = \int f_1(x)g_1(x){\rm{d}}x +\int f_2(x)g_2(x){\rm{d}}x.$$
Here is what I have tried so far. By imposing the orthogonality condition we can obtain the following system. \begin{cases} \langle G^{\perp}, \nabla Q_1 \rangle=0\implies \langle G,\nabla Q_1 \rangle = \langle g_1\cdot\nabla Q_1 ,\nabla Q_1 \rangle + \langle g_1\cdot\nabla Q_1,\nabla Q_2 \rangle \\ \langle G^{\perp}, \nabla Q_2 \rangle=0\implies \langle G,\nabla Q_2 \rangle = \langle g_2\cdot\nabla Q_2 ,\nabla Q_1\rangle + \langle g_2\cdot \nabla Q_2,\nabla Q_2 \rangle \end{cases} I am not sure how to get analytic expressions for $g_1$ and $g_2.$ Any ideas, comments will be much appreciated.