Global strong solutions to radial symmetric compressible Navier–Stokes equations bounded by a free surface

ABSTRACT

In this paper, we consider the two-dimensional isentropic compressible Navier–Stokes equations bounded by a free surface that is under the surface tension and constant exterior pressure. We establish the existence of global strong solution for arbitrary large spherical initial data with initial density away from the vacuum in the case the viscosity coefficients satisfy $\mu \left(\rho \right)=2\mu ,\lambda \left(\rho \right)={\rho }^{\beta },\beta >1$. In particular, we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive.

KEYWORDS:

• Compressible Navier–Stokes equations
• free boundary value problem
• global strong solution

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 76N15
• 35Q30

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Xingwei Zhang  http://orcid.org/0000-0003-0066-8767

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant no. 11701323] and the Doctoral Starting Foundation of Quzhou University [no. BSYJ201708].