I'm struggling to work out some simple formalism. I'm reading something that says if the quadruple $(v_i(x_i,t),p(x_i,t))$ solves the navier stokes equations so does the quadruple $(v_i^c(x,t),p^c(x,t))$ given by $(v_i(x_i-c_it,t)+c_i,p(x_i-c_it,t))$. I've assumed here $i$ ranges from 1 to 3. For the divergence condition, I work out the following. Letting $x_i' = x_i-c_it$ we have:
$$\sum_i \frac{\partial v^c_i}{\partial x'_i} = \sum_i \frac{\partial v^c_i}{\partial x_i}\frac{\partial x_i}{\partial x'_i} = \sum_i \frac{\partial v^c_i}{\partial x_i} = \sum_i \frac{\partial v_i(x_i-c_it,t)+c_i}{\partial x_i}$$
I want to conclude $$\sum_i \frac{\partial v^c_i}{\partial x'_i} = 0$$ but I'm getting stuck. I probably need to use $$\sum_i \frac{\partial v_i(x,t)}{\partial x_i} = 0$$ but I'm not sure how.