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: rho{t + (rho u)(x) = 0 in (0, T) x (0, 1) (rho u)(t) + (rho u(2))(x) + rho Phi(x) - (mu(rho)u(x))(x) + P-x = 0 in (0, T) x (0, 1) ((delta(Phi(x))(2) + 1/Phi(x))(2) + delta)(2-p/2) Phi(x))(x) = 4 pi g(rho - 1/vertical bar Omega vertical bar integral(Omega) rho dx) in (0, T) x (0, 1) Here, the Phi is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.