Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

We consider strong solutions to the initial boundary value problems for the isentropic compressible Navier–Stokes equations in one dimension: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho\left\{\begin{array}{lll} t+(\rho u)_x=0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, {\rm in}\,(0,T)\times(0,1)\\ (\rho u )_t+(\rho u^2)_x+\rho \Phi_x-(\mu( \rho )u_x)_x+P_x=0\quad\quad {\rm in}\,(0,T)\times(0,1) \\\left(\left(\frac{\delta(\Phi_x)^2\,+\,1}{(\Phi_x)^2\,+\,\delta}\right)^{\frac{2-p}{2}}\Phi_x\right)_x=4\pi g(\rho-\frac{1}{|\Omega|}\int\nolimits_\Omega \rho dx\,\,\,\, )\quad\,\, {\rm in}\,(0,T)\times(0,1)\end{array}\right.$$\end{document}Here, the Φ is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.