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The global existence of smooth solutions to the vacuum free boundary problemwith physical singularity of compressible Euler equations with damping andgravity is proved in space dimensions $n=1, 2, 3$, for the initial data beingsmall perturbations of the stationary solution. Moreover, the exponential decayof the velocity is obtained for $n=1, 2, 3$. The exponentially fast convergenceof the density and vacuum boundary to those of the stationary solution is shownfor $n=1$, and it is proved for $n=2, 3$ that they stay close to those of thestationary solution if they do so initially. The proof is based on the weightedestimates of both hyperbolic and parabolic types with weights capturing thesingular behavior of higher-order normal derivatives near vacuum states,exploring the balance between the physical singularity which pushes the vacuumboundary outwards and the effect of gravity which pulls it inwards, and thedissipation of the frictional damping. The results obtained in this paper arethe first ones on the global existence of solutions to the vacuum free boundaryproblems of inviscid compressible fluids with the non-expanding backgroundsolutions. Exponentially fast convergence when the vacuum state is involveddiscovered in this paper is a new feature of the problem studied.

Global Solution to the Vacuum Free Boundary Problem with Physical Singularity of Compressible Euler Equations with Damping and Gravity