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

By Raphaël Danchin, Francesco Fanelli. 2011

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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 ∞ , r s embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Holder spaces C 1 , α and of the endpoint Besov space B ∞ , 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.

Subfields: Applied Mathematics, Mathematical Physics

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