Analytical solutions to the 2D compressible Navier-Stokes equations with density-dependent viscosity coefficients

In this paper, we considered a class of analytical, rotational, and self-similar solutions to the 2D compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients. For the isentropic case $k>0,\gamma =\phi >1$, we provided the formula of self-similar analytical solutions and proved the well-posedness and the large time behavior for the corresponding generalized Emden equation. It is interesting to see that the different effects of rotation and pressure were revealed. Compared with the irrotational and pressureless case, when the free boundary $a(t)$ increases linearly or sub-linearly in time, we can find some classes of solutions with linear growth by taking the pressure effect or the swirl effect into account. In this sense, rotation or pressure effects may accelerate the growth of the boundary. Finally, we gave some examples of blow-up solutions and used a new method to prove the results.