Well-posedness for the Prandtl system without analyticity or monotonicity

Gérard-Varet, David; Masmoudi, Nader

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1273-1325.

Il a longtemps été supposé que l'équation de Prandtl n'est bien posée que sous l'hypothèse de monotonie d'Oleinik, ou pour des données analytiques. Nous montrons qu'elle est en fait localement bien posée pour des données appartenant à la classe Gevrey $7/4$ en la variable $x$. Nous améliorons ainsi le résultat classique d'existence locale de solutions analytiques en la variable $x$ (classe Gevrey 1). La preuve repose sur de nouvelles estimations, faisant appel à des fonctionnelles d'énergie non-quadratiques.

It has been thought for a while that the Prandtl system is only well-posed under the Oleinik monotonicity assumption or under an analyticity assumption. We show that the Prandtl system is actually locally well-posed for data that belong to the Gevrey class $7/4$ in the horizontal variable $x$. Our result improves the classical local well-posedness result for data that are analytic in $x$ (that is Gevrey class 1). The proof uses new estimates, based on non-quadratic energy functionals.

Classification : 35Q30, 35Q31, 35Q35.
Keywords: Boundary layer, Prandtl equation, Navier-Stokes equation, Gevrey spaces.
Mot clés : Couche limite, équation de Prandtl, équation de Navier-Stokes, espaces de Gevrey.

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