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

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.

Keywords

• global solution
• isothermal system
• friction terms
• viscosity-flux approximation
• compensated compactness