A Local Discontinuous Galerkin approximation for the $p$-Navier-Stokes system, Part II: Convergence rates for the velocity

In the present paper, we prove convergence rates for the Local Discontinuous Galerkin (LDG) approximation, proposed in Part I of the paper, for systems of $p$-Navier-Stokes type and $p$-Stokes type with $p\in (2,\mathrm{\infty })$. The convergence rates are optimal for linear ansatz functions. The results are supported by numerical experiments.