  The Kuramoto model is a mean-field model for the synchronisation behaviour of
oscillators, which exhibits Landau damping. In a recent work, the nonlinear
stability of a class of spatially inhomogeneous stationary states was shown
under the assumption of analytic regularity. This paper proves the nonlinear
Landau damping under the assumption of Sobolev regularity. The weaker
regularity required the construction of a different more robust bootstrap
argument, which focuses on the nonlinear Volterra equation of the order
parameter.
