  We propose in this work new systems of equations which we call $p$-Euler
equations and $p$-Navier-Stokes equations. $p$-Euler equations are derived as
the Euler-Lagrange equations for the action represented by the Benamou-Brenier
characterization of Wasserstein-$p$ distances, with incompressibility
constraint. $p$-Euler equations have similar structures with the usual Euler
equations but the `momentum' is the signed ($p-1$)-th power of the velocity. In
the 2D case, the $p$-Euler equations have streamfunction-vorticity formulation,
where the vorticity is given by the $p$-Laplacian of the streamfunction. By
adding diffusion presented by $\gamma$-Laplacian of the velocity, we obtain
what we call $p$-Navier-Stokes equations. If $\gamma=p$, the {\it a priori}
energy estimates for the velocity and momentum have dual symmetries. Using
these energy estimates and a time-shift estimate, we show the global existence
of weak solutions for the $p$-Navier-Stokes equations in $\mathbb{R}^d$ for
$\gamma=p$ and $p\ge d\ge 2$ through a compactness criterion.
