Three-phase Muskat problem: uniform lifespan with respect to the width of the strip between interfaces

We consider the three-phase Muskat problem with different densities and the same viscosities. The lifespan of the solutions with respect to the width of the strip between interfaces is studied. Indeed, the interfaces are parameterized by the graph of two functions $f(x,t)$ and $g(x,t)$ and we impose that $\mathrm{\mid }\mathrm{\mid }f(\cdot ,0)-g(\cdot ,0)\mathrm{\mid }{\mathrm{\mid }}_{L^{\mathrm{\infty }}}\le C\sigma$ and ${\inf }_x\mathrm{\mid }f(\cdot ,0)-g(\cdot ,0)\mathrm{\mid }\ge c\sigma \mathrm{.}$ It is shown, under stronger assumption on $f(x,0)$ and $g(x,0)$, local existence independent of the parameter $\sigma$ (with $\sigma$ small enough). In order to prove such a result, we need to work in analytic spaces.