Geodesic
Solvability of a~steady boundary value problem for a~model system of equations of a~barotropic motion of a~mixture of viscous compressible fluids
Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 3, pp. 55-67.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a boundary value problem for a model system of equations describing a steady barotropic motion of a homogeneous mixture of viscous compressible fluids in a bounded three-dimensional domain. An existence theorem is proved for weak solutions to the problem without constraints on the structure of the total viscosity matrix except the standard requirements of positive definiteness.
Keywords: existence theorem, steady boundary value problem, homogeneous mixture with two velocities
Mots-clés : viscous compressible fluid, effective viscous flux. 
@article{SJIM_2016_19_3_a4,
     author = {D. A. Prokudin and M. V. Krayushkina},
     title = {Solvability of a~steady boundary value problem for a~model system of equations of a~barotropic motion of a~mixture of viscous compressible fluids},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {55--67},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2016_19_3_a4/}
}
TY  - JOUR
AU  - D. A. Prokudin
AU  - M. V. Krayushkina
TI  - Solvability of a~steady boundary value problem for a~model system of equations of a~barotropic motion of a~mixture of viscous compressible fluids
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2016
SP  - 55
EP  - 67
VL  - 19
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2016_19_3_a4/
LA  - ru
ID  - SJIM_2016_19_3_a4
ER  - 
%0 Journal Article
%A D. A. Prokudin
%A M. V. Krayushkina
%T Solvability of a~steady boundary value problem for a~model system of equations of a~barotropic motion of a~mixture of viscous compressible fluids
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2016
%P 55-67
%V 19
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2016_19_3_a4/
%G ru
%F SJIM_2016_19_3_a4
D. A. Prokudin; M. V. Krayushkina. Solvability of a~steady boundary value problem for a~model system of equations of a~barotropic motion of a~mixture of viscous compressible fluids. Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 3, pp. 55-67. http://geodesic.mathdoc.fr/item/SJIM_2016_19_3_a4/

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

[2] Rajagopal K. L., Tao L., Mechanics of Mixtures, World Scientific Publ., Singapore, 1995 | MR | Zbl

[3] Mamontov A. E., Prokudin D. A., “Viscous compressible multi–fluids: modeling and multi-D existence”, Meth. Appl. Anal., 20:2 (2013), 179–195 | MR

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

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

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

[8] Kucher N. A., Mamontov A. E., Prokudin D. A., “Statsionarnye resheniya uravnenii dinamiki smesei vyazkikh szhimaemykh teploprovodnykh zhidkostei”, Sib. mat. zhurn., 53:6 (2012), 1338–1353 | MR | Zbl

[9] Mamontov A. E., Prokudin D. A., “Razreshimost statsionarnoi kraevoi zadachi dlya uravnenii dvizheniya odnotemperaturnoi smesi vyazkikh szhimaemykh teploprovodnykh zhidkostei”, Izv. RAN. Ser. mat., 78:3 (2014), 135–160 | DOI | MR | Zbl

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

[11] Feireisl E., Novotny A., Singular Limits in Thermodynamics of Viscous Fluids, Birkhauser, Basel, 2009 | MR | Zbl

[12] Novotny A., Straskraba I., Introduction to the Mathematical Theory of Compressible Flow, Univ. Press, Oxford, 2004 | MR | Zbl

[13] Plotnikov P. I., Sokolovski Zh., “Statsionarnye resheniya uravnenii Nave–Stoksa dlya dvukhatomnykh gazov”, Uspekhi mat. nauk, 62:3 (2007), 117–148 | DOI | MR | Zbl

[14] Bogovskii M. E., “O reshenii nekotorykh zadach vektornogo analiza, svyazannykh s operatorami div i grad”, Trudy seminara S. L. Soboleva, 1, 1980, 5–40 | MR

[15] Lions P.-L., Mathematical Topics in Fluid Mechanics, v. 2, Compressible Models, Oxford Univ. Press, NY, 1998 | MR | Zbl

[16] Agmon S., Douglis A., Nirenberg L., “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

[17] Agmon S., Douglis A., Nirenberg L., “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

[18] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR

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

[20] Tartar L., “Compensated compactness and applications to partial differential equations”, Nonlinear Analysis and Mech. Heriot–Watt Symp., v. IV, Res. Notes Math., 39, Pitman Publ., Boston, 1979, 136–212 | MR