Affine Solutions to the Compressible Euler Equations With Time‐Dependent Damping and Free Boundary

We consider the isentropic compressible Euler equations with time‐dependent damping and vacuum free boundary. The existence theory of affine solutions for the damped Euler equations was established by deriving an second‐order nonlinear ordinary differential equations (ODEs), and providing an asymptotic analysis of the corresponding matrix system. Our results show that the damping effect will influence the long‐time behavior of solutions. More specifically, for , both the diameter and volume of viscous fluids expand to infinity as time goes to infinity, and the algebraic expansion rate is not bigger than that of Euler equations. However, for , the fluid region possesses a finite upper bound, which is different from the algebraic expansion of the fluid region for the Euler equations. For the spherically symmetric case (a special affine solution), a more accurate expansion rate of the free boundary was obtained, where the free boundary will grow linearly in time for , sub‐linearly in time for , and has a finite upper bound for .