Density-dependent indecompressible viscous fluids in critical spaces. (English) Zbl 1050.76013

The author studies the initial value problem \[ \begin{aligned} \partial_{t} \rho + \text{div} \, \rho u &= 0, \qquad \partial_{t}(\rho u) + \text{div} \, (\rho u \otimes u) - \mu \triangle u + \nabla \Pi = \rho f,\\ \text{div} \, u &= 0, \qquad (\rho, u)| _{t=0} = (\rho_{0}, u_{0}), \end{aligned} \] in the whole space \(\mathbb{R}^{N}\), \((N \geq 2)\). The well-known results by Fujita and Kato for the constant density are generalized to the case when the initial density \(\rho_{0}\) is close to a constant. Moreover, local well-posedness is found for large initial velocity \(u_{0}\), and global well-posedness is established for initial velocity which is small with respect to a function of viscosity.


76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI