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

Huicheng Yin; Lin Zhang

doi:10.3934/dcds.2018045

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2018, Volume 38, Issue 3: 1063-1102. Doi: 10.3934/dcds.2018045

This issue Previous Article Dynamics in dimension zero A survey Next Article Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces

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

Huicheng Yin^1, , and
Lin Zhang^2,

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
^* Corresponding author: Huicheng Yin

Received: November 30, 2016

Revised: August 31, 2017

Published: June 2018

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.

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Abstract

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.

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Mathematics Subject Classification: 35L70, 35L65, 35L67, 76N15.

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Figure 1. A viscous fluid in a 3-D expanding ball

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