Global well-posedness and large-time behavior for anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation

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Abstract

In this paper, we consider the Cauchy problem for the two-dimensional anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation, where the density at infinity is in the state $\widetilde{\rho}>0$. We obtain the global well-posedness of strong solutions and the large-time behavior of the velocity field under the condition that $\left\|\sqrt{\rho_{0}} \mathbf{u}_{0}\right\|_{L^{2}}$ is sufficiently small. It is worth noting that our results include the case of vacuum.

Keywords

inhomogeneous Navier-Stokes equations / global strong solutions / partial dissipation / decay rates / vacuum

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