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On the well-posedness of the incompressible density-dependent Euler equations in the \(L^p\) framework. (English) Zbl 1192.35137
The author investigates the well-posedness for the density-dependent Euler equations in the whole space. Local-in-time results for the Cauchy problem are proven pertaining to data in the Besov spaces embedded in the set of Lipschitz functions, including the borderline case \(B^{p,1\frac{N}{p}+1}(\mathbb R^N)\) . A continuation criterion of Beale-Kato-Majda type for arbitrary time is also proven. The approach of Mr. Danchin is not restricted to the \(L^2\) framework or to small perturbations of a constant density state. The improvement was obtained by a new a priori estimate in Besov spaces for an elliptic equation with nonconstant coefficients.

MSC:
35Q31 Euler equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B45 A priori estimates in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
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