Abstract
In this paper, we investigate the Navier–Stokes equations describing the motion of a compressible viscous fluid confined to a thin domain varOmega _{varepsilon }=I_{varepsilon }times (0, 1), I_{ varepsilon }=(0, varepsilon )subset mathbb{R}. We show that the strong solutions in the 2D domain converge to the classical solutions of the limit 1D Navier–Stokes system as varepsilon to 0.
Highlights
The fluid is confined to a bounded physical domain Ωε ⊂ R2, on the boundary of which we impose the complete slip boundary conditions uε · n|∂Ωε = 0, S(∇xuε) · n × n|∂Ωε = 0, (1.2)
For three-dimensional system, Bella, Feireisl and Novotný in [1] considered the motion of a compressible viscous fluid confined to a cavity shaped as a thin rod Ωε = εQ × (0, 1), Q ⊂ R2, they showed that the weak solutions in the 3D domain converge to solutions of the limit 1D Navier–Stokes system as ε → 0
Motivated by [1, 5] and [2], our main purpose in this paper is to show that the strong solution of 2D compressible Navier–Stokes system confined to a thin domain Ωε = (0, ) × (0, 1) converge to the classical solution of the 1D Navier–Stokes system on (0, 1) as ε → 0
Summary
The fluid is confined to a bounded physical domain Ωε ⊂ R2, on the boundary of which we impose the complete slip boundary conditions uε · n|∂Ωε = 0, S(∇xuε) · n × n|∂Ωε = 0, (1.2) Under suitable conditions on the initial data it is natural to expect that the strong solution (ρε, uε) of (1.1)–(1.2) on Ωε tends, as ε → 0, to a classical solution (ρ, u) of the 1D system on (0, 1):
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