A generalised comparison principle for the Monge-Ampère equation and the pressure in 2D fluid flows (Q1701507)

This paper deals with the generalized comparison principle (Theorem 7) for the Monge-Ampère equation in a non-convex domain \(\Omega\). The case of strictly convex domains \(\Omega\) was first studied by \textit{J. Rauch} and \textit{B. A. Taylor} [Rocky Mt. J. Math. 7, 345--364 (1977; Zbl 0367.35025)]. As a consequence of Theorem 7, the author deduces in Corollary 8 bounds from above and below on solutions to the Monge-Ampère equation with sign-changing right-hand side \(f\). This way, for even \(n\), \(\Omega\) a bounded open set in \(\mathbb{R}^n\) and \(f\geqq 0\), \(f\not\equiv 0\), the equation equipped with constant Dirichlet data has no \(H^2(\Omega)\) solution. Due to the connection between the 2D Navier-Stokes equations and the Monge-Ampère equation, the pressure \(p\) in a 2D Navier-Stokes equation on a bounded domain cannot satisfy \(\Delta_xp\leq 0\), \(\Delta_x p\not\equiv 0\) at any \(t>0\).