## 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.

### MSC:

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

### Keywords:

well-posedness; initial value problem
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