  We study the existence of weak solutions to the $p$-Navier-Stokes equations
with a symmetric $p$-Laplacian on bounded domains. We construct a particular
Schauder basis in $W_0^{1,p}(\Omega)$ with divergence free constraint and prove
existence of weak solutions using the Galerkin approximation via this basis.
Meanwhile, in the proof, we establish a chain rule for the $L^p$ norm of the
weak solutions, which fixes a gap in our previous work. The equality of energy
dissipation is also established for the weak solutions considered.
