# American Institute of Mathematical Sciences

March  2018, 38(3): 1063-1102. doi: 10.3934/dcds.2018045

## The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations

 1 School of Mathematical Sciences, Jiangsu Provincial Key Laboratory, for Numerical Simulationof Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China 2 Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China

* Corresponding author: Huicheng Yin

Received  December 2016 Revised  September 2017 Published  December 2017

Fund Project: The authors were supported by the NSFC (No.11571177, No.11731007) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In this paper, as the second part of our three papers, we will confirm this physical phenomenon for the compressible viscid fluids by obtaining the exact lower and upper bound on the density function.

Citation: Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
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##### References:
A viscous fluid in a 3-D expanding ball
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