The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients
The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients
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DOI：
10.3934/krm.2008.1.313
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发表时间：
2008-05
影响因子：
1
通讯作者：
Q. Jiu;Z. Xin
中科院分区：
文献类型：
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作者：
Q. Jiu;Z. Xin
本文研究了粘性系数与密度相关的一维可压缩N-S方程的柯西问题。这里考虑了两种情况，第一，假设初始密度在整个实线上是可积的。第二，初始密度与正定密度的偏差在整个实线上是可积的。证明了在这两种情况下，柯西问题的弱解在时间上是整体存在的，并研究了这种弱解的大时间渐近性态。特别地，对于第二种情况，给出了真空消失和解的爆破现象，并证明了真空态消失后，整体弱解成为唯一的强解。初始真空是允许的，其结果适用于浅水一维圣维南模型。
This paper concerns with Cauchy problems for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficients. Two cases are considered here, first, the initial density is assumed to be integrable on the whole real line. Second, the deviation of the initial density from a positive constant density is integrable on the whole real line. It is proved that for both cases, weak solutions for the Cauchy problem exist globally in time and the large time asymptotic behavior of such weak solutions are studied. In particular, for the second case, the phenomena of vanishing of vacuum and blow-up of the solutions are presented, and it is also shown that after the vanishing of vacuum states, the globally weak solution becomes a unique strong one. The initial vacuum is permitted and the results apply to the one-dimensional Saint-Venant model for shallow water.