Revista Matemática Iberoamericana


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Volume 23, Issue 2, 2007, pp. 537–586
DOI: 10.4171/RMI/505

Published online: 2007-08-31

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

Hammadi Abidi[1]

(1) Université de Rennes I, Rennes, France

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.]).

Keywords: Inhomogenous fluid, existence, uniqueness

Abidi Hammadi: Equation de Navier-Stokes avec densité et viscosité variables dans l’espace critique. Rev. Mat. Iberoamericana 23 (2007), 537-586. doi: 10.4171/RMI/505