On the Cauchy Problem of 3D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity and vacuum. We establish some key a priori exponential decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence of strong solutions in the whole three-dimensional space, provided that the initial velocity is suitably small in the $\dot H^β$-norm for some $β\in(1/2,1].$ Note that this result is proved without any smallness conditions on the initial density. Moreover, the density can contain vacuum states and even have compact support initially.

No reviews yet.

No comments yet.