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We prove the global existence and uniqueness of smooth solutions to theone-dimensional barotropic Navier-Stokes system with degenerate viscosity$\mu(\rho)=\rho^\alpha$. We establish that the smooth solutions have possiblytwo different far-fields, and the initial density remains positive globally intime, for the initial data satisfying the same conditions. In addition, ourresult works for any $\alpha>0$, i.e., for a large class of degenerateviscosities. In particular, our models include the viscous shallow waterequations. This extends the result of Constantin-Drivas-Nguyen-Pasqualotto\cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case wheresmooth solutions connect possibly two different limits at the infinity on thewhole space.

Global smooth solutions for 1D barotropic Navier-Stokes equations with a large class of degenerate viscosities