I want to prove global existence of a wave map on a Robertson Walker spacetime $V= I \times S$, where $I=[0, \infty) \subset \mathbb{R}$ with metric $$ g=-dt^2+R^2 \sigma $$ where $\sigma$ is a Riemannian metric on $S$ and $R$ a smooth function on $I$.
The wave equation is defined as
$$ \text{tr}_g (\nabla^2 u)=0$$
The sources I use are mainly "Global Wave Maps on Robertson–Walker Spacetimes" by YVONNE CHOQUET-BRUHAT and "The Cauchy Problem in General Relativity" by Hans Ringström.
In the paper by Choquet-Bruhat, it is shown that, provided we have a local solution to the wave equation on $V$, be obtain a global solution by constructing energy estimates and showing, that we are able to control the speed of propagation of the energy. However, local existence is not shown in the paper.
To show local existence, I want to use the source by Ringström. Locally, the wave equation $$ \text{tr}_g (\nabla^2 u)=0$$ is a non linear wave equation on $\mathbb{R}^{n+1}$. Ringström proves, that nonlinear wave equations have unique local solutions.
So what I obtain is the following: I can cover $S$ by open sets $W_i$ in $V$, s.t. I have a solution $u_i$ to the wave equation on each $W_i$. What I would like to do next is to glue the solutions together to obtain a global solution, but I don't really know how to do that.
As far as I see, the uniqueness of the solution of the non linear wave equation on $\mathbb{R}^{n+1}$ doesn't help me here, since it it depending on the chart I am using, right?
So what I would like to use is an energy estimate, which is contructed in the paper by Choquet-Bruhat, which is of the following form:
$$\int\limits_{\{t\} \times S} \vert d_t u \vert^2+ R^{-2} \vert du \vert^2 \ dV=: e_t \leq e_0R(0)^2R^{-2}$$
If I could use this statement, I could obtain uniqueness at the intersections $W_i \cap W_j$, but in the energy estimate, we use that $\partial S = \emptyset$ which can not be guaranteed, if we exchange $\{t\} \times S$ by $W_i \cap W_j$ in the energy estimate.
I know that Ringström also provides a proof for the existence of a global wave equation on globally hyperbolic Lorentzian manifolds. But he is using a different kind of proof than Choquet Bruhat and does not provide a proof for local existence so if I would use his proof to show local existence, I could as well stick with his proof for global existence but I want to use the Choquet Bruhat paper.
I tried to find other sources that show local existence but in these sources, it often says something like "this is a known result from ..." for example Leray theory.
Does anybody has a hint for a source, where this is proven or can give me an idea how to prove this?
I only have one idea: The only part in the proof for the energy estimate stated above, where it is crucial that the boundary is empty is an integral over a vector field divergence on the right hand side of the equation, which, by Stokes, is zero. If I could somehow restrict the $W_i$, s.t. I could show that this vector field divergence is negative on the intersections, I could get the same estimate as above.