A generalised comparison principle for the Monge-Amp\`ere equation and the pressure in 2D fluid flows

Abstract

We extend the generalised comparison principle for the Monge-Amp\`ere equation due to Rauch & Taylor (Rocky Mountain J. Math. 7, 1977) to nonconvex domains. From the generalised comparison principle we deduce bounds (from above and below) on solutions of the Monge-Amp\`ere equation with sign-changing right-hand side. As a consequence, if the right-hand side is nonpositive (and does not vanish almost everywhere) then the equation equipped with constant boundary condition has no solutions. In particular, due to a connection between the two-dimensional Navier-Stokes equations and the Monge-Amp\`ere equation, the pressure $p$ in 2D Navier-Stokes equations on a bounded domain cannot satisfy $\Delta p \leq 0$ in $\Omega $ unless $\Delta p \equiv 0$ (at any fixed time). As a result at any time $t>0$ there exists $z\in \Omega $ such that $\Delta p (z,t) =0$.