Regularity of solutions to 1D compressible Navier–Stokes–Allen–Cahn system

Abstract

In this paper, we investigate the compressible Navier–Stokes equations coupled with an Allen–Cahn phase field description, which describes two-phase patterns in a flowing liquid including phase transformations. Our purpose is to establish the regularity of solutions in $H^i\left(\right[0,1\left]\right)(i=2,4)$ to the 1D Navier–Stokes–Allen–Cahn system for initial data ${\rho }_0$ without vacuum states.

Keywords:

• Navier–Stokes equations
• Allen–Cahn equation
• regularity

MSC 2010:

• 35Q30
• 35Q35
• 76N10
• 76D05

Acknowledgments

The author would like to thank the referees for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by NNSF of China [grant number 11801133] and the grant from the Key Research Projects of He'nan Higher Education Institutions [grant number 18A110038].