Revista Matemática Iberoamericana

Equation de Navier-Stokes avec densité et viscosité variables dans l'espace critique

Hammadi Abidi

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Abstract

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.]).

Article information

Source
Rev. Mat. Iberoamericana, Volume 23, Number 2 (2007), 537-586.

Dates
First available in Project Euclid: 26 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1190831221

Mathematical Reviews number (MathSciNet)
MR2371437

Zentralblatt MATH identifier
1175.35099

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]

Keywords
inhomogeneous fluid existence uniqueness

Citation

Abidi, Hammadi. Equation de Navier-Stokes avec densité et viscosité variables dans l'espace critique. Rev. Mat. Iberoamericana 23 (2007), no. 2, 537--586. https://projecteuclid.org/euclid.rmi/1190831221


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