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EXISTENCE RESULT FOR HEAT-CONDUCTING VISCOUS INCOMPRESSIBLE FLUIDS WITH VACUUM
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 Title & Authors
EXISTENCE RESULT FOR HEAT-CONDUCTING VISCOUS INCOMPRESSIBLE FLUIDS WITH VACUUM
Cho, Yong-Geun; Kim, Hyun-Seok;
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 Abstract
The Navier-Stokes system for heat-conducting incompressible fluids is studied in a domain . The viscosity, heat conduction coefficients and specific heat at constant volume are allowed to depend smoothly on density and temperature. We prove local existence of the unique strong solution, provided the initial data satisfy a natural compatibility condition. For the strong regularity, we do not assume the positivity of initial density; it may vanish in an open subset (vacuum) of or decay at infinity when is unbounded.
 Keywords
heat-conducting incompressible Navier-Stokes equations;strong solutions;vacuum;
 Language
English
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 References
1.
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975

2.
H. Amann, Heat-conducting incompressible viscous fluids, Navier-Stokes equations and related nonlinear problems (Funchal, 1994), 231-243, Plenum, New York, 1995

3.
M. E. Bogovskii, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1037-1040

4.
J. L. Boldrini, M. A. Rojas-Medar, and E. Fern'andez-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl. (9) 82 (2003), no. 11, 1499-1525 crossref(new window)

5.
Y. Cho, H. J. Choe, and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl. (9) 83 (2004), no. 2, 243-275 crossref(new window)

6.
Y. Cho and H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Anal. 59 (2004), no. 4, 465-489 crossref(new window)

7.
Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations 228 (2006), no. 2, 377-411 crossref(new window)

8.
H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations 28 (2003), no. 5-6, 1183-1201 crossref(new window)

9.
E. Fern'andez Cara and F. Guillen, The existence of nonhomogeneous, viscous and incompressible flow in unbounded domains, Comm. Partial Differential Equations 17 (1992), no. 7-8, 1253-1265

10.
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I., Linearized steady problems. Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994

11.
M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math. 330 (1982), 173-214

12.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, 224. Springer-Verlag, Berlin, 1983

13.
J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), no. 5, 639-681 crossref(new window)

14.
S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity, Tokyo J. Math. 22 (1999), no. 1, 17-42 crossref(new window)

15.
J. U. Kim, Weak solutions of an initial-boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal. 18 (1987), no. 1, 89-96 crossref(new window)

16.
H. Kozono and T. Ogawa, Some Lp estimate for the exterior Stokes flow and an application to the nonstationary Navier-Stokes equations, Indiana Univ. Math. J. 41 (1992), no. 3, 789-808 crossref(new window)

17.
H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J. 40 (1991), no. 1, 1-27 crossref(new window)

18.
O. Ladyzhenskaya and V. A. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Boundary value problems of mathematical physics, and related questions of the theory of functions, 8. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52 (1975), 52-109, 218-219

19.
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1., Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996

20.
H. Okamoto, On the equation of nonstationary stratified fluid motion: uniqueness and existence of the solutions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 615-643

21.
M. Padula, On the existence and uniqueness of nonhomogeneous motions in exterior domains, Math. Z. 203 (1990), no. 4, 581-604 crossref(new window)

22.
R. Salvi, The equations of viscous incompressible nonhomogeneous fluids: on the existence and regularity, J. Austral. Math. Soc. Ser. B 33 (1991), no. 1, 94-110 crossref(new window)

23.
J. Simon, Ecoulement d'um fluide non homogene avec une densite initiale s'annulant, C. R. Acad. Sci. Paris Ser. A-B 287 (1978), no. 15, A1009-A1012

24.
J. Simon, Compact sets in the space $L^p$(0; T;B), Ann. Mat. Pura Appl. (4) 146 (1987), 65-96 crossref(new window)

25.
J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal. 21 (1990), no. 5, 1093-1117 crossref(new window)

26.
V. A. Solonnikov, Solvability of the initial boundary value problem for the equation of a viscous compressible fluid, J. Sov. Math. 14 (1980), 1120-1133 crossref(new window)

27.
R. Temam, Navier-Stokes Equations, Theory and numerical analysis. With an appendix by F. Thomasset. Third edition. Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam, 1984