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

Huicheng Yin,,Department Of Mathematics And Ims, Nanjing University, Nanjing 210093, China ,Lin Zhang

doi:10.3934/dcds.2018045

Huicheng Yin, ,Department Of Mathematics And Ims, Nanjing University, Nanjing 210093, China + Show 1 more

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https://doi.org/10.3934/dcds.2018045

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Journal: Discrete & Continuous Dynamical Systems - A	Publication Date: Dec 1, 2017
Citations: 2	License type: cc-by

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.

Highlights

In this paper, we study the global existence and stability of a smooth compressible viscous fluid in a 3-D expanding ball
The global existence of classical solution was first obtained in [22, 23] for the initial data close to a non-vacuum state and these results were generalized in other settings, for example in [5, 27]
We prove the following local existence result on the problem (10)-(11) with (12)

Summary

Introduction

We study the global existence and stability of a smooth compressible viscous fluid in a 3-D expanding ball. Compressible Navier-Stokes equations, expanding ball, weighted energy estimates, global existence, vacuum state. The global existence of classical solution was first obtained in [22, 23] for the initial data close to a non-vacuum state and these results were generalized in other settings, for example in [5, 27]. For the case when the initial density may vanish in some region, under the smallness assumption on the total energy, the authors in [14] established the global existence and uniqueness of classical solutions. The authors in [7] obtained the global existence of weak solution to the compressible barotropic Navier-Stokes system in a time dependent domain with slip boundary condition. To solve (10)-(11) with (12), we will study the folllowing linearized problem for (τ, y) ∈ [0, T ] × S0: Φτ

ΦR V R

Dw H2 dτ

H3 t 0 τs Dv

H3 τs v