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

Takayuki Kobayashi; Wojciech Zajączkowski

Applicationes Mathematicae (1999)

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

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 + α , 1 + α / 2 ( Ω × ℝ + ) and the density belongs to H 1 + α , 1 / 2 + α / 2 ( Ω × ℝ + ) , 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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  1. [1] R. A. Adams, Sobolev Spaces, Acad. Press, New York, 1975. 
  2. [2] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1975 (in Russian); English transl.: Scripta Series in Mathematics, Winston and Halsted Press, 1979. 
  3. [3] M. Burnat and W. M. Zajączkowski, On local motion of compressible barotropic viscous fluid with the boundary slip condition, to appear. Zbl0916.35086
  4. [4] K. K. Golovkin, On equivalent norms for fractional spaces, Trudy Mat. Inst. Steklov. 66 (1962), 364-383 (in Russian). 
  5. [5] L. Landau and E. Lifschitz, Mechanics of Continuum Media, Nauka, Moscow, 1954 (in Russian); English transl.: Pergamon Press, Oxford, 1959; new edition: Hydrodynamics, Nauka, Moscow, 1986 (in Russian). 
  6. [6] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat conductive fluids, Proc. Japan Acad. Ser. A 55 (1979), 337-342. Zbl0447.76053
  7. [7] A. Matsumura and T. Nishida, The initial value problems for the equations of motion of compressible viscous and heat conductive gases, J. Math. Kyoto Univ. 20 (1980), 67-104. Zbl0429.76040
  8. [8] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids, Comm. Math. Phys. 89 (1983), 445-464. Zbl0543.76099
  9. [9] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of general fluids, in: Computing Methods in Applied Sciences and Engineering, V, R. Glowinski and J. L. Lions (eds.), North-Holland, Amsterdam, 1982, 389-406. Zbl0505.76083
  10. [10] J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik, Bd. VIII/1, Springer, Berlin, 1959. 
  11. [11] V. A. Solonnikov and V. E. Shchadilov, On a boundary value problem for a stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 125 (1973), 196-210 (in Russian); English transl.: Proc. Steklov Inst. Math. 125 (1973), 186-199. 
  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
  15. [15] A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa (4) 10 (1983), 607-647. Zbl0542.35062
  16. [16] A. Valli and W. M. Zajączkowski, Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys. 103 (1986), 259-296. Zbl0611.76082

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