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 \(\beta_\sigma(f_0')\) which encapsulates local monotonicity and slope, we identify a new class of initial data within \(W^{ 1,\infty}(\mathbb{ R}^d)\). This includes scenarios where the product of the maximal and minimal slopes is large, thereby guaranteeing the local existence of a classical solution.
Categories: math.AP