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The well-posedness issue for the density-dependent in endpoint Besov spaces

Abstract : This work is the continuation of the recent paper (Danchin, 2010) [9] devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type B(infinity.r)(s) embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Holder spaces C(1 ,alpha) and of the endpoint Besov space B(infinity.1)(1). For such data and under the non-vacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale. Kato and Majda (1984) [2]. In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations. (C) 2011 Elsevier Masson SAS. All rights reserved.
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Raphaël Danchin, Francesco Fanelli. The well-posedness issue for the density-dependent in endpoint Besov spaces. Journal de Mathématiques Pures et Appliquées, Elsevier, 2011, 96 (3), pp.253--278. ⟨10.1016/j.matpur.2011.04.005⟩. ⟨hal-00692810⟩

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