Solubility of a~stationary boundary-value problem for the equations of motion of a~one-temperature mixture of viscous compressible heat-conducting fluids

A. E. Mamontov ; D. A. Prokudin

Izvestiya. Mathematics , Tome 78 (2014) no. 3, pp. 554-579.

Résumé

We consider a boundary-value problem describing the stationary motion of a two-component mixture of viscous compressible heat-conducting fluids in a bounded domain. We make no simplifying assumptions except for postulating the coincidence of phase temperatures (which is physically justified in certain situations), that is, we retain all summands in equations that are a natural generalization of the Navier–Stokes–Fourier model of the motion of a one-component medium. We prove the existence of weak generalized solutions of the problem.

Détail

Keywords: existence theorem, stationary boundary-value problem, viscous compressible heat-conducting fluid, homogeneous two-speed mixture, effective viscous flow.

Bibliographie

[1] R. I. Nigmatulin, Dinamika mnogofaznykh sred, ch. 1, Nauka, M., 1987

[2] K. L. Rajagopal, L. Tao, Mechanics of mixtures, Ser. Adv. Math. Appl. Sci., 35, World Scientific, River Edge, NJ, 1995 | MR | Zbl

[3] A. E. Mamontov, D. A. Prokudin, “Viscous compressible multi-fluids: modeling and multi-D existence”, Methods Appl. Anal., 20:2 (2013), 179–196 | DOI | MR | Zbl

[4] J. Frehse, S. Goj, J. Málek, “On a Stokes-like system for mixtures of fluids”, SIAM J. Math. Anal., 36:4 (2005), 1259–1281 | DOI | MR | Zbl

[5] J. Frehse, S. Goj, J. Málek, “A uniqueness result for a model for mixtures in the absence of external forces and interaction momentum”, Appl. Math., 50:6 (2005), 527–541 | DOI | MR | Zbl

[6] J. Frehse, W. Weigant, “On quasi-stationary models of mixtures of compressible fluids”, Appl. Math., 53:4 (2008), 319–345 | DOI | MR | Zbl

[7] N. A. Kucher, D. A. Prokudin, “Analiz razreshimosti kraevoi zadachi dlya uravnenii smesei vyazkikh szhimaemykh zhidkostei”, Vestn. Kemer. gos. un-ta, 1:45 (2011), 32–38

[8] N. A. Kucher, A. E. Mamontov, D. A. Prokudin, “Stationary solutions to the equations of dynamics of mixtures of heat-conductive compressible viscous fluids”, Siberian Math. J., 53:6 (2012), 1075–1088 | DOI | MR | Zbl

[9] A. A. Papin, Kraevye zadachi dvukhfaznoi filtratsii, AltGU, Barnaul, 2009

[10] A. N. Petrov, “Korrektnost nachalno-kraevykh zadach dlya odnomernykh uravnenii vzaimopronikayuschego dvizheniya sovershennykh gazov”, Dinamika neodnorodnoi zhidkosti, Dinamika sploshnoi sredy, v. 56, In-t gidrodinamiki, Novosibirsk, 1982, 105–121

[11] P.-L. Lions, Mathematical topics in fluid mechanics, v. 2, Oxford Lecture Ser. Math. Appl., 10, Compressible Models, Oxford University Press, New York, 1998 | MR | Zbl

[12] E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Ser. Math. Appl., 26, Oxford University Press, Oxford, 2004 | MR | Zbl

[13] A. Novotný, I. Straškraba, Introduction to the mathematical theory of compressible flow, Oxford Lecture Ser. Math. Appl., 27, Oxford University Press, Oxford, 2004 | MR | Zbl

[14] E. Feireisl, A. Novotný, Singular limits in thermodynamics of viscous fluids, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2009 | MR | Zbl

[15] P. Plotnikov, J. Sokolowski, Compressible Navier–Stokes equations. Theory and shape optimization, IMPAN Monogr. Mat. (N. S.), 73, Birkhäuser, Basel, 2012 | MR | Zbl

[16] O. V. Voinov, V. V. Pukhnachev, “Termokapillyarnoe dvizhenie v gazozhidkostnoi smesi”, Prikladnaya mekhanika i tekhnicheskaya fizika, 21:5 (1980), 38–45

[17] B. T. Zhumagulov, V. N. Monakhov, Gidrodinamika neftedobychi, KazgosINTI, Almaty, 2001

[18] S. K. Gard, J. W. Prichett, “Dynamics of gas-fluidized beds”, J. Appl. Phys., 46:10 (1975), 4493–4500 | DOI

[19] A. Novotný, M. Pokorný, “Steady compressible Navier–Stokes–Fourier system for monoatomic gas and its generalizations”, J. Differential Equations, 251:1 (2011), 270–315 | DOI | MR | Zbl

[20] P. B. Mucha, M. Pokorný, “On the steady compressible Navier–Stokes–Fourier system”, Comm. Math. Phys., 288:1 (2009), 349–377 | DOI | MR | Zbl

[21] P. B. Mucha, M. Pokorný, “Weak solutions to equations of steady compressible heat conducting fluids”, Math. Models Methods Appl. Sci., 20:5 (2010), 785–813 | DOI | MR | Zbl

[22] R. J. DiPerna, P.-L. Lions, “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math., 98:3 (1989), 511–547 | DOI | MR | Zbl

[23] M. E. Bogovskii, “O reshenii nekotorykh zadach vektornogo analiza, svyazannykh s operatorami $\operatorname{div}$ i $\operatorname{grad}$”, Tr. sem. S. L. Soboleva, 1, In-t matem. SO AN SSSR, Novosibirsk, 1980, 5–40

[24] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I”, Comm. Pure Appl. Math., 12:4 (1959), 623–727 | DOI | MR | Zbl

[25] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II”, Comm. Pure Appl. Math., 17:1 (1964), 35–92 | DOI | MR | Zbl

[26] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[27] R. R. Coifman, Y. Meyer, “On commutators of singular integrals and bilinear singular integrals”, Trans. Amer. Math. Soc., 212, 1975, 315–331 | DOI | MR | Zbl

[28] L. Tartar, “Compensated compactness and applications to partial differential equations”, Nonlinear analysis and mechanics, Res. Notes in Math., 39, Pitman, Boston, MA, 1979, 136–212 | MR | Zbl

[29] I. Ekeland, R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications, 1, North-Holland, Amsterdam–Oxford; Elsevier, New York, 1976 | MR | MR | Zbl | Zbl