Analytical solutions to some one-dimensional compressible fluid equations

In this paper, we construct some self-similar analytical solutions for some one-dimensional compressible fluid equations. For the isentropic Euler equations with heat conduction, we present an analytical solution with the temperature decaying in time at the rate of $\mathcal{O}((1+t{)}^{-1})$ for the free boundary problem, provide an analytical solution for the Cauchy problem, and investigate its blowup and decay phenomena. We also construct some analytical solutions for the isothermal Euler equations with heat conduction. Moreover, we give an exact solution to the isentropic Euler equations with time-dependent damping and construct an analytical solution to the Navier-Stokes equations with density-dependent viscosity for $\gamma =3$, respectively, where $\gamma$ is the adiabatic exponent.