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Existence and uniqueness for an inhomogeneous fluid. (Existence et unicité pour un fluide inhomogène.) (French. Abridged English version) Zbl 1099.35092

Summary: Recently R. Danchin [ Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 6, 1311–1334 (2003; Zbl 1050.76013)] showed the existence and uniqueness for an inhomogeneous fluid in the homogeneous Besov space \(\dot B_{21}^{\frac N2} (\mathbb R^N)\times \dot B_{21}^{-1+\frac N2} (\mathbb R^N)\), under the condition that \(\rho_0-1\) is small in \(\dot B_{2\infty}^{\frac N2}\cap L^\infty\) if \(2<N\), in \(\dot B_{21}^{\frac N2}\) if \(N=2\). In this note, one shows that the condition \(\|\rho_0-1\|_{L^\infty}\ll 1\) is sufficient to have the existence and uniqueness.

MSC:

35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 1050.76013
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References:

[1] H. Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Revista Matemática Iberoamericana, à paraître; H. Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Revista Matemática Iberoamericana, à paraître · Zbl 1175.35099
[2] Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14, 209-246 (1981) · Zbl 0495.35024
[3] Chemin, J.-Y., Fluides Parfaits Incompressibles, Astérisque, vol. 230 (1995) · Zbl 0829.76003
[4] Danchin, R., Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133, 1311-1334 (2003) · Zbl 1050.76013
[5] Danchin, R., Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations. Comm. Partial Differential Equations, Comm. Partial Differential Equations, 27, 11-12, 2531-2532 (2002), Erratum
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