On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid

Ewa Zadrzyńska; Wojciech M. Zajączkowski

Annales Polonici Mathematici (1996)

  • Volume: 63, Issue: 3, page 199-221
  • ISSN: 0066-2216

Abstract

top
We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.

How to cite

top

Ewa Zadrzyńska, and Wojciech M. Zajączkowski. "On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid." Annales Polonici Mathematici 63.3 (1996): 199-221. <http://eudml.org/doc/262879>.

@article{EwaZadrzyńska1996,
abstract = {We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.},
author = {Ewa Zadrzyńska, Wojciech M. Zajączkowski},
journal = {Annales Polonici Mathematici},
keywords = {viscous compressible heat conducting fluid; global existence; free boundary problem; motion of a viscous compressible heat conducting fluid; existence of global-in-time solutions},
language = {eng},
number = {3},
pages = {199-221},
title = {On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid},
url = {http://eudml.org/doc/262879},
volume = {63},
year = {1996},
}

TY - JOUR
AU - Ewa Zadrzyńska
AU - Wojciech M. Zajączkowski
TI - On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid
JO - Annales Polonici Mathematici
PY - 1996
VL - 63
IS - 3
SP - 199
EP - 221
AB - We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.
LA - eng
KW - viscous compressible heat conducting fluid; global existence; free boundary problem; motion of a viscous compressible heat conducting fluid; existence of global-in-time solutions
UR - http://eudml.org/doc/262879
ER -

References

top
  1. [1] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free boundary, Comm. Pure Appl. Math. 31 (1980), 359-392. Zbl0464.76028
  2. [2] J. T. Beale, Large time regularity of viscous surface waves, Arch. Rational Mech. Anal. 84 (1984), 307-352. Zbl0545.76029
  3. [3] 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. 
  4. [4] L. Landau and E. Lifschitz, Mechanics of Continuum Media, Nauka, Moscow, 1984 (in Russian); English transl.: Pergamon Press, Oxford, 1959; new edition: Hydrodynamics, Nauka, Moscow, 1986 (in Russian). 
  5. [5] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), 67-104. Zbl0429.76040
  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 boundary value problem for the equations of motion of compressible viscous and heat-conductive fluid, preprint of Univ. of Wisconsin, MRC Technical Summary Report no. 2237 (1981). Zbl0543.76099
  8. [8] 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, R. Glowinski and J. L. Lions (eds.), North-Holland, Amsterdam, 1982, 389-406. Zbl0505.76083
  9. [9] 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
  10. [10] V. A. Solonnikov, A priori estimates for parabolic equations of second order, Trudy Mat. Inst. Steklov 70 (1964), 133-212 (in Russian). Zbl0168.08202
  11. [11] V. A. Solonnikov, On an unsteady flow of a finite mass of a liquid bounded by a free surface, Zap. Nauchn. Sem. LOMI 152 (1986), 137-157 (in Russian); English transl.: J. Soviet Math. 40 (1988), 672-686. Zbl0614.76026
  12. [12] V. A. Solonnikov, Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. LOMI 140 (1984), 179-186 (in Russian); English transl.: J. Soviet Math. 32 (1986), 223-238. Zbl0551.76022
  13. [13] V. A. Solonnikov, On an unsteady motion of an isolated volume of a viscous incompressible fluid, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 1065-1087 (in Russian). 
  14. [14] V. A. Solonnikov and A. Tani, Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid, preprint of Paderborn University. Zbl0786.35106
  15. [15] 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
  16. [16] 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
  17. [17] 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
  18. [18] 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
  19. [19] 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
  20. [20] W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Dissertationes Math. 324 (1993). Zbl0771.76059
  21. [21] W. M. Zajączkowski, On local motion of a compressible barotropic viscous fluid bounded by a free surface, in: Partial Differential Equations, Banach Center Publ. 27, Part 2, Inst. Math., Polish Acad. Sci., 1992, 511-553. Zbl0791.35105
  22. [22] W. M. Zajączkowski, Existence of local solutions for free boundary problems for viscous compressible barotropic fluids, Ann. Polon. Math. 60 (1995), 255-287. Zbl0923.35134
  23. [23] W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous capillary fluid bounded by a free surface, SIAM J. Math. Anal. 25 (1994), 1-84. Zbl0813.35086

Citations in EuDML Documents

top
  1. Ewa Zadrzyńska, Wojciech Zajączkowski, Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting fluids
  2. Ewa Zadrzyńska, Wojciech M. Zajączkowski, On a differential inequality for a viscous compressible heat conducting capillary fluid bounded by a free surface
  3. Alfred Wagner, Nonstationary Marangoni convection
  4. Ewa Zadrzyńska, Wojciech Zajączkowski, On nonstationary motion of a fixed mass of a viscous compressible barotropic fluid bounded by a free boundary

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.