The Maximal Regularity of Nonlocal Parabolic Monge–Ampère Equations and Its Monotonicity in the Whole Space

The Monge–Ampère operator, as a nonlinear operator embedded in parabolic differential equations, complicates the demonstration of maximal regularity for these equations. This research uses the Riesz fractional derivative to connect the Monge–Ampère operator with the fractional Laplacian operator. It is then possible to seek the maximal regularity of the parabolic Monge–Ampère equations by following an approach similar to that used for finding the maximal regularity of the parabolic fractional Laplacian operator. The maximal regularity of nonlocal parabolic Monge–Ampère equations guarantees the existence of solutions in the whole space. Based on these conditions, a modified sliding method, an enhancement of the moving planes method, is employed to establish the monotonicity property of the solutions for the nonlocal parabolic Monge–Ampère equations in the whole space.