On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework

Raphaël Danchin (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in the set of Lipschitz functions, including the borderline case $B^{\frac Np+1}_{p,1}(\R^N).$ A continuation criterion in the spirit of the celebrated one by Beale-Kato-Majda for the classical Euler equations, is also proved. In contrast with the previous work dedicated to this system in the whole space, our approach is not restricted to the $L^2$ framework or to small perturbations of a constant density state: we just need the density to be bounded away from zero. The key to that improvement is a new a priori estimate in Besov spaces for an elliptic equation with nonconstant coefficients.

Mots clés: Incompressible Euler equations; transport equation; nonlinear analysis; elliptic estimates; Fourier analysis; Besov spaces

Domaines: Equations aux dérivées partielles [math.AP]

Raphaël Danchin. On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework. Journal of Differential Equations, 2010, 248 (8), pp.2130-2170. ⟨hal-00406595⟩