Double periodic viscous flows in infinite space-periodic pipes

We study the motion of an incompressible fluid in an $n+1$-dimensional infinite pipe $\,\La\,$ with an $L$-periodic shape in the $z=x_{n+1}$ direction. We set $\,x=(x_1,x_2,\cdots,x_{n})$, and $z=x_{n+1}$. We denote by $\Sigma_z$ the cross section of the pipe at the level $z\,,$ and by $v_z$ the $n+1$ component of the velocity. Fluid motion is described by the evolution Stokes or Navier-Stokes equations together with the non-slip boundary condition $\bv=\,0\,$. We look for solutions $\bv(x,z,t)$ with a given, arbitrary, $T-$time periodic total flux $\,\int_{\Sigma_z} \,v_z(x,z,t)\,dx=g(t)\,,$ which should be simultaneously $T$-periodic with respect to time and $L$-periodic with respect to $z\,.$ We prove existence and uniqueness of the solution to the above problems. The results extend those proved in reference \cite{B-05}, where the cross sections were independent of $z$. The argument is presented through a sequence of steps. We start by considering the linear, stationary, $z-$periodic Stokes problem. Then we study the double periodic evolution Stokes equations, which is the heart of the matter. Finally, we end with the extension to the full Navier-Stokes equations.