On the well-posedness of 3-D inhomogeneous incompressible Navier-Stokes equations with variable viscosity
A 3D Navier-Stokes incompressible fluid with non-constant density and variable viscosity is studied. The aim of the paper is to improve some previous well-posednesss results in critical Besov spaces (that means the corresponding norm are invariant under some specific transformationms). Some smallness conditions on the bounded means oscillation (BOM) norm of the value \(a_0=1/ \rho_0-1\) are imposed, where \(\rho_0\) is the initial density. With this condition, the Coifman-Rochberg-Weiss theorem and a new comutator estimates of integral type are used to obtain pressure estimates in terms of \(a\). Then a unique local solution exists for small enough \(a_0\). If the viscosity is constant and if the Besov norms of \(a_0\) and initial velocity are small enough, a global solution is obtained. Moreover, estimates of the global solution are given, in terms of the initial conditions. Interpolation inequalities and embedding relations in Besov spaces, the Littlewood-Paley theory, the Bopny's decomposition are used as mathematical tools. The authors specify that ``This result can be also regarded partially as an improvement'' of [\textit{R. Danchin} and \textit{P. B. Mucha}, Commun. Pure Appl. Math. 65, No. 10, 1458--1480 (2012; Zbl 1247.35088)].
scientific article; zbMATH DE number 6807003
scientific article