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The present paper considers a homogeneous bubble inside an unboundedpolytropic compressible liquid with viscosity. The system is governed by theNavier-Stokes equation with free boundary which is determined by the kinematicand dynamic boundary conditions on the bubble-liquid interface. The globalexistence of solution is proved, and the $\dot{H}^1$ asymptotic stability ofthe spherical equilibrium in terms of viscous damping together with a explicitdecay rate is given in bare energy methods.

The viscous damping of three dimensional spherical gas bubble inside unbounded compressible liquid