Previous |  Up |  Next

Article

Keywords:
compressible viscous fluids; miscible mixtures; quasi-stationary
Summary:
We consider mixtures of compressible viscous fluids consisting of two miscible species. In contrast to the theory of non-homogeneous incompressible fluids where one has only one velocity field, here we have two densities and two velocity fields assigned to each species of the fluid. We obtain global classical solutions for quasi-stationary Stokes-like system with interaction term.
References:
[1] Bernardi, Ch., Pironneau, O.: On the shallow water equations at low Reynolds number. Commun. Partial Differential Equations 16 (1991), 59-104. DOI 10.1080/03605309108820752 | MR 1096834 | Zbl 0723.76033
[2] Feireisl, E.: Compressible Navier-Stokes equations with a non-monotone pressure law. J. Differ. Equations 184 (2002), 97-108. DOI 10.1006/jdeq.2001.4137 | MR 1929148 | Zbl 1012.76079
[3] Feireisl, E.: Dynamics of Compressible Flow. Oxford University Press Oxford (2003).
[4] Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3 (2001), 358-392. DOI 10.1007/PL00000976 | MR 1867887
[5] Frehse, J., Goj, S., Málek, J.: On a Stokes-like system for mixtures of fluids. SIAM J. Math. Anal. 36 (2005), 1259-1281 (electronic). DOI 10.1137/S0036141003433425 | MR 2139449 | Zbl 1084.35057
[6] Frehse, J., Goj, S., Málek, J.: A uniqueness result for a model for mixtures in the absence of external forces and interaction momentum. Appl. Math. 50 (2005), 527-541. DOI 10.1007/s10492-005-0035-x | MR 2181024 | Zbl 1099.35079
[7] Gagliardo, E.: Ulteriori proprietà di alcune classi di funzioni in più variabili. Ric. Mat. 8 (1959), 24-51. MR 0109295 | Zbl 0199.44701
[8] Goj, S.: Analysis for mixtures of fluids. Bonner Math. Schriften. Dissertation Universität Bonn, Math. Inst. (2005), http://bib.math.uni-bonn.de/pdf2/BMS-375.pdf MR 2205590
[9] Golovkin, K. K.: On imbedding theorems. Soviet Math. Dokl. 1 (1960), 998-1001. MR 0121640 | Zbl 0104.33102
[10] Haupt, P.: Continuum Mechanics Theory of Materials, 2nd ed. Advanced Texts in Physics. Springer Berlin (2002). MR 2011110
[11] Il'in, V. P.: On theorems of ``imbedding''. Trudy Mat. Inst. Steklov. 53 (1959), 359-386. MR 0112930
[12] Kazhikhov, A. V.: Resolution of boundary value problems for nonhomogeneous viscous fluids. Dokl. Akad. Nauk 216 (1974), 1008-1010 Russian.
[13] Kazhikhov, A. V.: The equations of potential flow of compressible viscous fluid at low Reynolds number. Acta Appl. Math. 37 (1994), 77-81. DOI 10.1007/BF00995131 | MR 1308747 | Zbl 0815.35083
[14] Ladyzhenskaya, O. A., Solonnikov, V. A.: On the unique solvability of the initial value problem for viscous incompressible inhomogeneous fluids. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (LOMI) 52 (1975), 52-109, 218-219 Russian. Zbl 0376.76021
[15] Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 1: Compressible Models. Oxford Science Publications. Clarendon Press Oxford (1996). MR 1422251
[16] Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models. Oxford Science Publications. Clarendon Press Oxford (1998). MR 1637634
[17] Mamontov, A. E.: Well-posedness of a quasistationary model of a viscous compressible fluid. Sib. Mat. Zh. 37 (1996), 1117-1131. DOI 10.1007/BF02110728 | MR 1643271 | Zbl 0884.76077
[18] Nouri, A., Poupaud, F., Demay, Y.: An existence theorem for the multi-fluid Stokes problem. Q. Appl. Math. 55 (1997), 421-435. MR 1466141 | Zbl 0882.35091
[19] Nouri, A., Poupaud, F.: An existence theorem for the multifluid Navier-Stokes problem. J. Differ. Equations 122 (1995), 71-88. DOI 10.1006/jdeq.1995.1139 | MR 1356130 | Zbl 0842.35079
[20] Rajagopal, K. R., Tao, L.: Mechanics of Mixtures. Series on Advances in Mathematics for Applied Sciences Vol. 35. World Scientific Publishing River Edge (1995). MR 1370661
[21] Solonnikov, V. A.: On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov 83 (1965), 3-163. MR 0211083 | Zbl 0164.12502
[22] Solonnikov, V. A.: On the solvability of the initial-boundary problem for the equations of motion of a viscous compressible fluid. Zap. Nauchn. Semin. Leningrad, Otd. Mat. Inst. Steklova (LOMI) 56 (1976), 128-142. MR 0481666
[23] Tani, A.: On the first initial-boundary value problem of compressible viscous fluid motion. Publ. Res. Inst. Math. Sci., Kyoto Univ. 13 (1977), 193-253. DOI 10.2977/prims/1195190106 | Zbl 0366.35070
[24] Vaigant, V. A., Kazhikhov, A. V.: Global solutions of equations of potential flows of a compressible viscous fluid for small Reynolds numbers. Differentsial'nye Uravneniya 30 (1994), 1010-1022. MR 1312722
[25] Yudovich, V. I.: Nichtstationäre Strömung einer idealen inkompressiblen Flüssigkeit. Zh. Vychisl. Mat. Fiz. 3 (1963), 1032-1066 Russian. Zbl 0129.19402
[26] Yudovich, V. I.: Linearization Method in the Hydrodynamic Stability Theory. Izdatel'stvo Rostovskogo Universiteta Rostov (1984), Russian. Zbl 0553.76038
Partner of
EuDML logo