Well-posedness of the 2D Muskat equation with initial data in different Lebesgue spaces

This article investigates the existence, uniqueness, and regularity of solutions to the Muskat equation describing the motion of two immiscible fluids with constant densities in an incompressible porous medium, where the velocity is governed by Darcy's law. The initial data and its first and second derivatives are assumed to belong to different Lebesgue spaces.