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August, 2007 Equation de Navier-Stokes avec densité et viscosité variables dans l'espace critique
Hammadi Abidi
Rev. Mat. Iberoamericana 23(2): 537-586 (August, 2007).


In this article, we show that the Navier-Stokes system with variable density and viscosity is locally well-posed in the Besov space $$ \dot B^{\frac{N}{p}}_{p\,1}(\R^N)\times\big(\dot B^{\frac{N}{p}-1}_{p\,1}(\R^N)\big)^N, $$ for $1 < p\leq N$ when the initial density approaches a strictly positive constant. This result generalizes the work by R. Danchin for the case where the viscosity is constant and $p=2$ (see [Danchin, R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1311-1334.]). Moreover, we prove existence and uniqueness in the Sobolev space\arriba{2} $$ H^{\frac{N}{2}+\alpha}(\R^N)\times\big(H^{\frac{N}{2}-1+\alpha}(\R^N)\big)^N $$ for $\alpha>0,$ generalizing R. Danchin's result for the case where viscosity is constant (see [Danchin, R.: Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353-386.]).


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Hammadi Abidi. "Equation de Navier-Stokes avec densité et viscosité variables dans l'espace critique." Rev. Mat. Iberoamericana 23 (2) 537 - 586, August, 2007.


Published: August, 2007
First available in Project Euclid: 26 September 2007

zbMATH: 1175.35099
MathSciNet: MR2371437

Primary: 35Q30
Secondary: 35B30 , 76D03 , 76D05

Keywords: existence , inhomogeneous fluid , uniqueness

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.23 • No. 2 • August, 2007
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