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

TL;DR

This paper extends a comparison principle for the Monge-Ampère equation to nonconvex domains, deriving bounds on solutions with sign-changing right-hand sides and applying these results to the pressure in 2D fluid flows, showing constraints on its Laplacian.

Contribution

It generalizes the comparison principle for the Monge-Ampère equation to nonconvex domains and links these results to properties of pressure in 2D Navier-Stokes equations.

Findings

Solutions with nonpositive right-hand side and constant boundary conditions do not exist.

The pressure in 2D Navier-Stokes cannot have Laplacian strictly negative or positive everywhere.

At any positive time, the Laplacian of pressure must be zero at some point in the domain.

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 $\Delta p (z,t) =0$.