JournalsrmiVol. 23, No. 2pp. 537–586

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

  • Hammadi Abidi

    Université de Rennes I, Rennes, France
Equation de Navier-Stokes avec densité et viscosité variables dans l’espace critique cover

Abstract

In this article, we show that the Navier-Stokes system with variable density and viscosity is locally well-posed in the Besov space

B˙p1Np(RN)×(B˙p1Np1(RN))N,\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<pN1 < 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=2p=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}

HN2+α(RN)×(HN21+α(RN))NH^{\frac{N}{2}+\alpha}(\R^N)\times\big(H^{\frac{N}{2}-1+\alpha}(\R^N)\big)^N

for α>0,\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.]).

Cite this article

Hammadi Abidi, Equation de Navier-Stokes avec densité et viscosité variables dans l’espace critique. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 537–586

DOI 10.4171/RMI/505