Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting fluids

Ewa Zadrzyńska; Wojciech Zajączkowski

Applicationes Mathematicae (1998)

  • Volume: 25, Issue: 2, page 179-220
  • ISSN: 1233-7234

Abstract

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The local existence and the uniqueness of solutions for equations describing the motion of viscous compressible heat-conducting fluids in a domain bounded by a free surface is proved. First, we prove the existence of solutions of some auxiliary problems by the Galerkin method and by regularization techniques. Next, we use the method of successive approximations to prove the local existence for the main problem.

How to cite

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Zadrzyńska, Ewa, and Zajączkowski, Wojciech. "Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting fluids." Applicationes Mathematicae 25.2 (1998): 179-220. <http://eudml.org/doc/219199>.

@article{Zadrzyńska1998,
abstract = {The local existence and the uniqueness of solutions for equations describing the motion of viscous compressible heat-conducting fluids in a domain bounded by a free surface is proved. First, we prove the existence of solutions of some auxiliary problems by the Galerkin method and by regularization techniques. Next, we use the method of successive approximations to prove the local existence for the main problem.},
author = {Zadrzyńska, Ewa, Zajączkowski, Wojciech},
journal = {Applicationes Mathematicae},
keywords = {free boundary; compressible viscous heat-conducting fluids; local existence; Lagrangian coordinates; auxiliary problem; Galerkin method; regularization techniques; Schauder-Tikhonov fixed point theorem; iterative procedure; uniqueness},
language = {eng},
number = {2},
pages = {179-220},
title = {Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting fluids},
url = {http://eudml.org/doc/219199},
volume = {25},
year = {1998},
}

TY - JOUR
AU - Zadrzyńska, Ewa
AU - Zajączkowski, Wojciech
TI - Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting fluids
JO - Applicationes Mathematicae
PY - 1998
VL - 25
IS - 2
SP - 179
EP - 220
AB - The local existence and the uniqueness of solutions for equations describing the motion of viscous compressible heat-conducting fluids in a domain bounded by a free surface is proved. First, we prove the existence of solutions of some auxiliary problems by the Galerkin method and by regularization techniques. Next, we use the method of successive approximations to prove the local existence for the main problem.
LA - eng
KW - free boundary; compressible viscous heat-conducting fluids; local existence; Lagrangian coordinates; auxiliary problem; Galerkin method; regularization techniques; Schauder-Tikhonov fixed point theorem; iterative procedure; uniqueness
UR - http://eudml.org/doc/219199
ER -

References

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  1. [1] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1986 (in Russian). 
  2. [2] B. Ducomet, Hydrodynamical models od gaseous stars, Rep. Math. Phys., to appear. Zbl0949.76071
  3. [3] L. Landau and E. Lifschitz, Hydrodynamics, Nauka, Moscow, 1986 (in Russian); English transl.: Fluid Mechanics, Pergamon Press, Oxford, 1987. 
  4. [4] P. Secchi, On the motion of gaseous stars in presence of radiation, Comm. Partial Differential Equations 15 (1990), 185-204. Zbl0708.35096
  5. [5] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Steklov. 83 (1965) (in Russian); English transl.: Proc. Steklov Inst. Math. 83 (1967). Zbl0164.12502
  6. [6] G. Ströhmer and W. M. Zajączkowski, Local existence of solutions of free boundary problem for the equations of compressible barotropic viscous self-gravitating fluids, to appear. Zbl1016.76065
  7. [7] E. Zadrzyńska and W. M. Zajączkowski, On local motion of a general compressible viscous heat conducting fluid bounded by a free surface, Ann. Polon. Math. 59 (1994), 133-170. Zbl0812.35102
  8. [8] E. Zadrzyńska and W. M. Zajączkowski, Conservation laws in free boundary problems for viscous compressible heat-conducting fluids, Bull. Polish Acad. Sci. Tech. Sci. 42 (1994), 197-207. Zbl0814.76075
  9. [9] E. Zadrzyńska and W. M. Zajączkowski, On a differential inequality for equations of a viscous compressible heat conducting fluid bounded by a free surface, Ann. Polon. Math. 61 (1995), 141-188. Zbl0833.35156
  10. [10] E. Zadrzyńska and W. M. Zajączkowski, On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid, ibid. 63 (1996), 199-221. Zbl0862.35147
  11. [11] E. Zadrzyńska and W. M. Zajączkowski, On global motion of a compressible viscous heat conducting fluid bounded by a free surface, Acta Appl. Math. 37 (1994), 221-231. Zbl0813.35130
  12. [12] E. Zadrzyńska and W. M. Zajączkowski, On nonstationary motion of a fixed mass of a viscous compressible barotropic fluid bounded by a free boundary, in preparation. Zbl0930.35141

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