  We study long-time existence and asymptotic behaviour for a class of
anisotropic, expanding curvature flows. For this we adapt new curvature
estimates, which were developed by Guan, Ren and Wang to treat some stationary
prescribed curvature problems. As an application we give a unified flow
approach to the existence of smooth, even $L_p$-Minkowski problems in
$\mathbb{R}^{n+1}$ for $p>-n-1.$
