A generalised comparison principle for the Monge–Ampère equation and the pressure in 2D fluid flows

Ożański, Wojciech S.

Comptes Rendus. Mathématique, Tome 356 (2018) no. 2, pp. 198-206.

Nous étendons aux domaines non convexes le principe de comparaison généralisé pour l'équation de Monge–Ampère, dû à Rauch et Taylor. Nous en déduisons des bornes (supérieure et inférieure) pour les solutions de l'équation de Monge–Ampère avec second membre changeant de signe. En conséquence, si le second membre est négatif ou nul (et ne s'annule pas presque partout), alors l'équation avec condition au bord constante n'a pas de solution. En particulier, en raison d'une relation entre les équations de Navier–Stokes en dimension 2 et l'équation de Monge–Ampère, la pression p dans les équations de Navier–Stokes de dimension 2 sur un domaine borné Ω satisfait Δp≤0 dans Ω, à moins que Δp≡0 (à tout temps donné). Il en résulte qu'à tout temps t>0, il existe z∈Ω tel que Δp(z,t)=0.

We extend the generalised comparison principle for the Monge–Ampère equation due to Rauch & Taylor (1977) [15] to nonconvex domains. From the generalised comparison principle, we deduce bounds (from above and below) on solutions to the Monge–Ampère 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 a constant boundary condition has no solutions. In particular, due to a connection between the two-dimensional Navier–Stokes equations and the Monge–Ampère equation, the pressure p in 2D Navier–Stokes equations on a bounded domain cannot satisfy Δp≤0 in Ω unless Δp≡0 (at any fixed time). As a result, at any time t>0 there exists z∈Ω such that Δp(z,t)=0.

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