Cauchy problem for isothermal system in a general nozzle with space-dependent friction

Yun-guang Lu; Xian-ting Wang; Richard De la cruz; Yun-guang Lu; Xian-ting Wang; Richard De la cruz

doi:10.3934/math.2021381

AIMS Mathematics

2021, Volume 6, Issue 6: 6482-6489. doi: 10.3934/math.2021381

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Research article

Cauchy problem for isothermal system in a general nozzle with space-dependent friction

1.
K. K. Chen Institute for Advanced Studies, Hangzhou Normal University, China
2.
Wuxi Institute of Technology, China
3.
School of Mathematics and Statistics, Universidad Pedagógica y Tecnológica de Colombia, Colombia

Received: 08 February 2021 Accepted: 09 April 2021 Published: 15 April 2021
MSC : 35L65, 76N10

In this paper, we study the Cauchy problem of the isothermal system in a general nozzle with space-dependent friction $ \alpha(x) $. First, by using the maximum principle, we obtain the uniform bound $ \rho^{\delta, \varepsilon, \tau} \le M $, $ |m^{\delta, \varepsilon, \tau}| \le M $, independent of the time, of the viscosity-flux approximation solutions; Second, by using the compensated compactness method coupled with the convergence framework given in ^[5], we prove that the limit, $ (\rho, m) $ of $ (\rho^{\delta, \varepsilon, \tau}, m^{\delta, \varepsilon, \tau}) $, as $ \varepsilon, \delta, \tau $ go to zero, is a uniformly bounded entropy solution.
- global solution,
- isothermal system,
- friction terms,
- viscosity-flux approximation,
- compensated compactness
Citation: Yun-guang Lu, Xian-ting Wang, Richard De la cruz. Cauchy problem for isothermal system in a general nozzle with space-dependent friction[J]. AIMS Mathematics, 2021, 6(6): 6482-6489. doi: 10.3934/math.2021381

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Abstract

In this paper, we study the Cauchy problem of the isothermal system in a general nozzle with space-dependent friction $ \alpha(x) $. First, by using the maximum principle, we obtain the uniform bound $ \rho^{\delta, \varepsilon, \tau} \le M $, $ |m^{\delta, \varepsilon, \tau}| \le M $, independent of the time, of the viscosity-flux approximation solutions; Second, by using the compensated compactness method coupled with the convergence framework given in ^[5], we prove that the limit, $ (\rho, m) $ of $ (\rho^{\delta, \varepsilon, \tau}, m^{\delta, \varepsilon, \tau}) $, as $ \varepsilon, \delta, \tau $ go to zero, is a uniformly bounded entropy solution.

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