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Navier-Stokes equation with variable density and viscosity in critical space. (Equation de Navier-Stokes avec densité et viscosité variables dans l’espace critique.) (French) Zbl 1175.35099
The author proves that the incompressible Navier-Stokes system with variable density \(\rho\) and density-dependent viscosity coefficient \(\mu(\rho)\) is well-posed in the homogeneous Besov space \[ \dot{B}^{N/p}_{p\;1}({\mathbb R}^N)\times \left(\dot{B}^{N/p-1}_{p\;1}({\mathbb R}^N)\right)^N, \] for \(1\leq p\leq N\), provided that \[ 0<\underline{\mu}\leq \mu(\rho)\;\;\text{and}\;\mu\in C^{\infty}. \] This result extends a previous one by R. Danchin [ Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 6, 1311–1334 (2003; Zbl 1050.76013)], valid for \(p=2\) and \(\mu=\text{const}\). A result of global existence and uniqueness is also proved in the Sobolev framework \[ H^{N/2+\alpha}({\mathbb R}^N)\times \left(H^{N/2-1+\alpha}({\mathbb R}^N)\right)^N, \] for any \(\alpha>0\).

MSC:
35Q30 Navier-Stokes equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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