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

Abstract

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 ${\dot{H}}^{\beta }$-norm for some $\beta \in (1/2,1].$ 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.

23 pages

References in corpus (4)

• Global Wellposeness for the 3D inhomogeneous incompressible Navier-Stokes equations
• On Local Strong Solutions to the Cauchy Problem of Two-Dimensional Density-Dependent Magnetohydrodynamic Equations with Vacuum
• Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Navier-Stokes Equations with Vacuum
• On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity

Cited by in corpus (6)

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• Exponential Stability of the Inhomogeneous Navier-Stokes-Vlasov System in Vacuum
• On the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes system with density-dependent viscosity
• Global Strong Solutions to the incompressible Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum in 3D Exterior Domains
• Global well-posedness and exponential decay to the Cauchy problem of nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum in ${\mathbb{R}}^2$
• Global Existence of Strong Solutions and Serrin-Type Blowup Criterion for 3D Combustion Model in Bounded Domains