Global Smooth Solutions for 1D Barotropic Navier–Stokes Equations with a Large Class of Degenerate Viscosities

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier–Stokes system with degenerate viscosity μ(ρ)=ρα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (\rho )=\rho ^\alpha $$\end{document}. We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document}, i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 37:145–180, 2020, Theorem 1.6) (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.