On the well-posedness of the incompressible density-dependent Euler equations in the LP framework

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-p+1(N/p+1) (R-N) A continuation criterion in the spirit of the celebrated one by Beale. Kato and Majda (1984) in [2] 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 fiom zero The key to that Improvement is a new a prion estimate in Besov spaces for an elliptic equation with nonconstant coefficients (C) 2009 Elsevier Inc All rights reserved