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In this paper, we study some model problem associated to the free boundaryvalue problem of the barotropic compressible Navier-Stokes equations in generalsmooth domain with taking surface tension into account. To obtain the maximal$L_p-L_q$ regularity property of the model problem, we prove the existence of$\mathcal{R}-$bounded operator families of the resolvent problem via Weis'theory on operator valued Fourier multipliers.

The $\mathcal{R}-$bounded operator families arising from the study of the barotropic compressible flows with free surface