Analytical solutions to the 1D compressible isothermal Navier-Stokes equations with density-dependent viscosity

In this paper, we construct a class of analytical solutions to the one-dimensional compressible isothermal Navier-Stokes equations with density-dependent viscosity in the real line R. Precisely, we take the pressure p(rho) = a(1)rho and the viscosity coefficient mu(rho) = a(2)rho with a(1), a(2) > 0. We show that the system has an exact solution with the initial data satisfying rho(0)(x) = e(x) and u(0)(x) = x. The large-time asymptotic behavior of the density is exhibited according to various a(1). The analytical solutions to the compressible isothermal Euler equations and the pressureless Euler equations are obtained as by-products.