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



Revista Matemática Iberoamericana

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

Hammadi Abidi

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

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.]).

Primary Subjects: 35Q30
Secondary Subjects: 35B30, 76D03, 76D05
Keywords: inhomogeneous fluid; existence; uniqueness

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