Global Existence And Exponential Stability Of Solutions In H～4 For The Compressible Navier-Stokes Equations With The Cylinder Symmetry

This paper is concerned with the compressible Navier-Stokes equations with the cylinder symmetry in R~3, due to the flows that a polytropic fluid between two circular coaxial cylinders. The three-dimensional equations in Eulerian coordinates can be written in Lagrangian coordinates as follows: T_t=(ru)x, u_t=r[v(ru)x-γθ/τ]_x+v~2/r, v_t=μr[(rv)x/τ]_x-uv/r, w_t=μr[(rw)x/τ]_x+μτw/r~2, C_vθ_t=k[r~2θx/r]x+1/τ[v(ru)_x-γθ](ru)_x+μ(rv)~2/τ+μr~2w_x~2/τ-2μ(u~2+v~2)_xIn this paper, based on the results in [24], we establish the global existence and exponential stability of solutions in H~4 for the equations when the initial total energy(which does not include the initial density) is sufficiently small. Moreover, the global existence and exponential stability of the classical solution can be also derived.