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(ρ) = a1ρ and the viscosity coefficient μ(ρ) = a2ρ with a1, a2 > 0. We show that the system has an exact solution with the initial data satisfying ρ0(x) = ex and u0(x) = x. The large-time asymptotic behavior of the density is exhibited according to various a1. The analytical solutions to the compressible isothermal Euler equations and the pressureless Euler equations are obtained as by-products.