# On local motion of a compressible barotropic viscous fluid bounded by a free surface

Banach Center Publications (1992)

- Volume: 27, Issue: 2, page 511-553
- ISSN: 0137-6934

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topZajączkowski, W.. "On local motion of a compressible barotropic viscous fluid bounded by a free surface." Banach Center Publications 27.2 (1992): 511-553. <http://eudml.org/doc/262843>.

@article{Zajączkowski1992,

abstract = {We consider the motion of a viscous compressible barotropic fluid in ℝ³ bounded by a free surface which is under constant exterior pressure, both with surface tension and without it. In the first case we prove local existence of solutions in anisotropic Hilbert spaces with noninteger derivatives. In the case without surface tension the anisotropic Sobolev spaces with integration exponent p > 3 are used to omit the coefficients which are increasing functions of 1/T, where T is the existence time.},

author = {Zajączkowski, W.},

journal = {Banach Center Publications},

keywords = {free boundary; compressible barotropic viscous fluid; local existence; anisotropic Sobolev spaces; surface tension; barotropic fluid; Hölder space; initial boundary value problem; viscous compressible barotropic fluid; bounded domain; successive approximations},

language = {eng},

number = {2},

pages = {511-553},

title = {On local motion of a compressible barotropic viscous fluid bounded by a free surface},

url = {http://eudml.org/doc/262843},

volume = {27},

year = {1992},

}

TY - JOUR

AU - Zajączkowski, W.

TI - On local motion of a compressible barotropic viscous fluid bounded by a free surface

JO - Banach Center Publications

PY - 1992

VL - 27

IS - 2

SP - 511

EP - 553

AB - We consider the motion of a viscous compressible barotropic fluid in ℝ³ bounded by a free surface which is under constant exterior pressure, both with surface tension and without it. In the first case we prove local existence of solutions in anisotropic Hilbert spaces with noninteger derivatives. In the case without surface tension the anisotropic Sobolev spaces with integration exponent p > 3 are used to omit the coefficients which are increasing functions of 1/T, where T is the existence time.

LA - eng

KW - free boundary; compressible barotropic viscous fluid; local existence; anisotropic Sobolev spaces; surface tension; barotropic fluid; Hölder space; initial boundary value problem; viscous compressible barotropic fluid; bounded domain; successive approximations

UR - http://eudml.org/doc/262843

ER -

## References

top- [1] R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.
- [2] G. Allain, Small-time existence for the Navier-Stokes equations with a free surface, Appl. Math. Optim. 16 (1987), 37-50.
- [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] 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] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), 481-524. Zbl0476.76068
- [6] T. Nishida, Equations of fluid dynamics-free surface problems, ibid. 39 (1986), 221-238.
- [7] P. Secchi and A. Valli, A free boundary problem for compressible viscous fluids, J. Reine Angew. Math. 341 (1983), 1-31. Zbl0502.76082
- [8] V. A. Solonnikov, On an initial-boundary value problem for the Stokes system which appears in free boundary problems, Trudy Mat. Inst. Steklov. 188 (1990), 150-188 (in Russian).
- [9] V. A. Solonnikov, On the solvability of the initial-boundary value problem for equations of motion of a viscous compressible fluid, Zap. Nauchn. Sem. LOMI 56 (1976), 128-142 (in Russian). Zbl0338.35078
- [10] V. A. Solonnikov, On boundary 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
- [11] V. A. Solonnikov, On an unsteady motion of an isolated volume of a viscous incompressible fluid, Izv. Akad. Nauk. SSSR Ser. Mat. 51 (5) (1987), 1065-1087 (in Russian).
- [12] V. A. Solonnikov, A priori estimates for second order parabolic equations, Trudy Mat. Inst. Steklov. 70 (1964), 133-212 (in Russian). Zbl0168.08202
- [13] V. A. Solonnikov and A. Tani, Free boundary problem for a viscous compressible flow with a surface tension, in: Constantine Carathéodory: An International Tribute, T. M. Rassias (ed.), World Scientific, 1991, 1270-1303. Zbl0752.35096
- [14] P. Weidemaier, Refinement of an ${L}^{p}$-estimate of Solonnikov for a parabolic equation of the second order with conormal boundary condition, Math. Z. 199 (1988), 589-604. Zbl0683.35038
- [15] W. M. Zajączkowski, On an initial-boundary value problem for the parabolic system which appears in free boundary problems for compressible Navier-Stokes equations, Dissertationes Math. 304 (1990).
- [16] W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, ibid., to appear. Zbl0813.35086
- [17] W. M. Zajączkowski, A priori estimates for solutions to noncharacteristic mixed problems to nonlinear symmetric hyperbolic systems of the first order with dissipation, Bull. Polish Acad. Sci. Math. 37 (1-6) (1989), 183-197. Zbl0770.35042
- [18] W. M. Zajączkowski, On nonstationary motion of compressible barotropic viscous capillary fluid bounded by a free surface, to appear. Zbl0813.35086

## Citations in EuDML Documents

top- Ewa Zadrzyńska, Wojciech M. Zajączkowski, On local motion of a general compressible viscous heat conducting fluid bounded by a free surface
- W. M. Zajączkowski, Existence of local solutions for free boundary problems for viscous compressible barotropic fluids
- Ewa Zadrzyńska, Wojciech M. Zajączkowski, On a differential inequality for equations of a viscous compressible heat conducting fluid bounded by a free surface
- Ewa Zadrzyńska, Wojciech M. Zajączkowski, On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid

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