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

## TL;DR

This paper proves the global existence and uniqueness of smooth solutions for the 1D barotropic Navier-Stokes equations with a broad class of degenerate viscosities, including models like viscous shallow water equations, extending previous periodic domain results.

## Contribution

It establishes the global well-posedness of smooth solutions with possibly different far-fields for all positive alpha, broadening the class of degenerate viscosities covered.

## Key findings

- Global existence and uniqueness of smooth solutions.
- Solutions can connect two different far-field states.
- Results apply to a large class of degenerate viscosities including shallow water models.

## Abstract

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier-Stokes system with degenerate viscosity $\mu(\rho)=\rho^\alpha$. 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 $\alpha>0$, 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-Drivas-Nguyen-Pasqualotto \cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.