Maximal models of torsors over a local field

Let $R$ be a discrete valuation ring, and $K$ its fraction field. In 1967, Raynaud initiated the notion of maximal $R$-model for torsors over $K$, and it was further developed by Lewin-M\'en\'egaux. In this paper, motivated by a conjectural ramification theory for infinitesimal torsors, we investigate this notion of maximal model in greater detail. We prove the maximality, the compatibility along inductions, and an existence result for group schemes of semi-direct products.