The Muskat problem with a large slope

In this paper, we establish local well-posedness results for the Muskat equation in any dimension using modulus of continuity techniques. By introducing a novel quantity ${\beta }_{\sigma }(f_0^{\mathrm{\prime }})$ which encapsulates local monotonicity and slope, we identify a new class of initial data within $W^{1,\mathrm{\infty }}({\mathbb{R}}^d)$. This includes scenarios where the product of the maximal and minimal slopes is large, thereby guaranteeing the local existence of a classical solution.