Abstract
This paper is concerned with the regularity of the global solutions in $H^{4}$ to the compressible isentropic Navier-Stokes equations with the cylinder symmetry in $R^{3}$ . Such a circular coaxial cylinder symmetric domain is an unbounded domain, but we assume that the corresponding solution depends only on one radial variable, r in $G=\{r\in R^{+},0< a\leq r\leq b<+\infty\}$ , in which the related domain G is a bounded domain. Some new ideas and more delicate estimates are introduced to prove these results.
Highlights
1 Introduction The compressible isentropic Navier-Stokes equations with density-dependent viscosity coefficients can be written for t > as ρt + div(ρU) =, ( . )
In this paper we establish the regularity of global solutions to the compressible NavierStokes equations with cylinder symmetry in R
For the cylindrically symmetric problem to the three-dimensional compressible Navier-Stokes equations, when the viscosity coefficients are both constants, the uniqueness of the weak solutions was proved in [, ], the global existence of isentropic compressible cylindrically symmetric solution was established in [ ]; this result was later generalized to the nonisentropic case in [ ]
Summary
For the cylindrically symmetric problem to the three-dimensional compressible Navier-Stokes equations, when the viscosity coefficients are both constants, the uniqueness of the weak solutions was proved in [ , ], the global existence of isentropic compressible cylindrically symmetric solution was established in [ ]; this result was later generalized to the nonisentropic case in [ ]. When viscosity coefficients μ and λ are density-dependent, Liu and Lian [ ] established the global existence and asymptotic behavior of cylindrically symmetric solutions, there is no result on the regularity for this system.
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