# On global motion of a compressible barotropic viscous fluid with boundary slip condition

• Volume: 26, Issue: 2, page 159-194
• ISSN: 1233-7234

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## Abstract

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Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ ${ℝ}^{3}$ with the boundary slip condition is proved. The solution is close to an equilibrium solution. The proof is based on the energy method. Moreover, in the ${L}_{2}$-approach the result is sharp (the regularity of the solution cannot be decreased) because the velocity belongs to ${H}^{2+\alpha ,1+\alpha /2}\left(\Omega ×{ℝ}_{+}\right)$ and the density belongs to ${H}^{1+\alpha ,1/2+\alpha /2}\left(\Omega ×{ℝ}_{+}\right)$, α ∈ (1/2,1).

## How to cite

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Kobayashi, Takayuki, and Zajączkowski, Wojciech. "On global motion of a compressible barotropic viscous fluid with boundary slip condition." Applicationes Mathematicae 26.2 (1999): 159-194. <http://eudml.org/doc/219232>.

@article{Kobayashi1999,
abstract = {Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ $ℝ^3$ with the boundary slip condition is proved. The solution is close to an equilibrium solution. The proof is based on the energy method. Moreover, in the $L_2$-approach the result is sharp (the regularity of the solution cannot be decreased) because the velocity belongs to $H^\{2+α,1+α/2\}(Ω × ℝ_+)$ and the density belongs to $H^\{1+α,1/2+α/2\}(Ω× ℝ_+)$, α ∈ (1/2,1).},
author = {Kobayashi, Takayuki, Zajączkowski, Wojciech},
journal = {Applicationes Mathematicae},
keywords = {Hilbert-Besov spaces; compressible barotropic viscous fluid; boundary slip condition; global existence; global-in-time existence of solutions; viscous compressible barotropic fluid; slip boundary condition; energy method},
language = {eng},
number = {2},
pages = {159-194},
title = {On global motion of a compressible barotropic viscous fluid with boundary slip condition},
url = {http://eudml.org/doc/219232},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Kobayashi, Takayuki
AU - Zajączkowski, Wojciech
TI - On global motion of a compressible barotropic viscous fluid with boundary slip condition
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 2
SP - 159
EP - 194
AB - Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ $ℝ^3$ with the boundary slip condition is proved. The solution is close to an equilibrium solution. The proof is based on the energy method. Moreover, in the $L_2$-approach the result is sharp (the regularity of the solution cannot be decreased) because the velocity belongs to $H^{2+α,1+α/2}(Ω × ℝ_+)$ and the density belongs to $H^{1+α,1/2+α/2}(Ω× ℝ_+)$, α ∈ (1/2,1).
LA - eng
KW - Hilbert-Besov spaces; compressible barotropic viscous fluid; boundary slip condition; global existence; global-in-time existence of solutions; viscous compressible barotropic fluid; slip boundary condition; energy method
UR - http://eudml.org/doc/219232
ER -

## References

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12. [12] V. A. Solonnikov and A. Tani, Evolution free boundary problem for equations of motion of a viscous compressible barotropic liquid, Zap. Nauchn. Sem. LOMI 182 (1990), 142-148; also in: Constantin Carathéodory: an International Tribute, M. Rassias (ed.), Vol. 2, World Sci., 1991, 1270-1303. Zbl0723.76026
13. [13] G. Ströhmer, About a certain class of parabolic-hyperbolic systems of differential equations, Analysis 9 (1989), 1-39. Zbl0685.35078
14. [14] G. Ströhmer, About compressible viscous fluid flow in a bounded domain, Pacific J. Math. 143 (1990), 359-375. Zbl0669.76089
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