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

By Raphaël Danchin. 2009

114 citations recorded by the source corpus.

Abstract 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 from zero. The key to that improvement is a new a priori estimate in Besov spaces for an elliptic equation with nonconstant coefficients.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8028f4cd8f24a51028ce26f4d0ebf16340067040

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Optimal Time-decay Estimates for the Compressible Navier–Stokes Equations in the Critical Lp Framework (2016, unknown): “): (A.6) kA(D)fkLa .λmkfkLa. An obvious consequence of (A.5) and (A.6) is that kDkfk ˙ Bs p,r ≈kfkB˙s+k p,r for all k ∈ N. We also need the following nonlinear generalization of (A.6) (see Lemma 8 in [8]): Proposition A.4. If SuppFf ⊂ {ξ ∈ Rd : R1λ ≤ |ξ| ≤ R2λ} then there exists c depending only on ”
The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces (2011, methodology): “This work is the continuation of a recent paper by the first author (see [9]) devoted to the density-dependent incompressible Euler equations: { ∂tρ + u · ∇ρ = 0, ρ(∂tu+ u · ∇u)+∇Π = ρf, divu= 0.”