On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

Peixin  Zhang; Jianwen Zhang; Junning  Zhao

doi:10.3934/dcds.2016.36.1085

Article Contents

2016, Volume 36, Issue 2: 1085-1103. Doi: 10.3934/dcds.2016.36.1085

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On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

1.
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021
2.
School of Mathematical Sciences, Xiamen University, Xiamen 361005

Received: February 28, 2014

Revised: October 31, 2014

Published: May 2017

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Abstract

In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.

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Mathematics Subject Classification: Primary: 76N10; Secondary: 35M10.

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