EXISTENCE RESULT FOR HEAT-CONDUCTING VISCOUS INCOMPRESSIBLE FLUIDS WITH VACUUM

Title & Authors
EXISTENCE RESULT FOR HEAT-CONDUCTING VISCOUS INCOMPRESSIBLE FLUIDS WITH VACUUM
Cho, Yong-Geun; Kim, Hyun-Seok;

Abstract
The Navier-Stokes system for heat-conducting incompressible fluids is studied in a domain $\small{{\Omega}{\subset}R^3}$. 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 $\small{{\Omega}}$ or decay at infinity when $\small{{\Omega}}$ is unbounded.
Keywords
heat-conducting incompressible Navier-Stokes equations;strong solutions;vacuum;
Language
English
Cited by
1.
Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Computers & Mathematics with Applications, 2011, 61, 9, 2742
2.
Invariant elliptic estimates, Journal of Mathematical Analysis and Applications, 2011, 382, 1, 162
3.
Regularity of 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity, Journal of Differential Equations, 2008, 245, 12, 3956
4.
Local existence of unique strong solution to non-isothermal model for incompressible nematic liquid crystals in 3D, Applied Mathematics and Computation, 2016, 290, 487
5.
Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients, Zeitschrift für angewandte Mathematik und Physik, 2016, 67, 5
6.
Global strong solutions for 3D viscous incompressible heat conducting magnetohydrodynamic flows with non-negative density, Journal of Mathematical Analysis and Applications, 2017, 446, 1, 707
7.
The global wellposedness of the 3D heat-conducting viscous incompressible fluids with bounded density, Nonlinear Analysis: Real World Applications, 2015, 22, 129
8.
Blow-up of viscous compressible reactive self-gravitating gas, Acta Mathematicae Applicatae Sinica, English Series, 2012, 28, 2, 401
9.
Elliptic estimates independent of domain expansion, Calculus of Variations and Partial Differential Equations, 2009, 34, 3, 321
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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