On a mathematical model of non-isothermal creeping flows of a fluid through a given domain

A. A. Domnich ; E. S. Baranovskii ; M. A. Artemov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 3, pp. 417-429.

Résumé

We study a mathematical model describing steady creeping flows of a non-uniformly heated incompressible fluid through a bounded 3D domain with locally Lipschitz boundary. The model under consideration is a system of second-order nonlinear partial differential equations with mixed boundary conditions. On in-flow and out-flow parts of the boundary the pressure, the temperature and the tangential component of the velocity field are prescribed, while on impermeable solid walls the no-slip condition and a Robin-type condition for the temperature are used. For this boundary-value problem, we introduce the concept of a weak solution (a pair “velocity–temperature”), which is defined as a solution to some system of integral equations. The main result of the work is a theorem on the existence of weak solutions in a subspace of the Cartesian product of two Sobolev's spaces. To prove this theorem, we give an operator interpretation of the boundary value problem, derive a priori estimates of solutions, and apply the Leray–Schauder fixed point theorem. Moreover, energy equalities are established for weak solutions.

Détail

Mots-clés : flux problem
Keywords: non-isothermal flows, creeping flows, mixed boundary conditions, weak solutions.

Bibliographie

[1] Krein S. G., Chan Tkhu Kha., “The problem of the flow of a non-uniformly heated viscous fluid”, U.S.S.R. Comput. Math. Math. Phys., 29:4 (1989), 127–131 | DOI | MR | Zbl

[2] Kovtunov D. A., “Solvability of the stationary heat convection problem for a high-viscosity fluid”, Differ. Equ., 45:1 (2009), 73–85 | DOI | Zbl

[3] Korotkii A. I., “Solution in weak sense of a boundary value problem describing thermal convection”, Trudy Inst. Mat. i Mekh. UrO RAN, 16:2 (2010), 121–132 (In Russian)

[4] Alekseev G. V., “Solvability of stationary problems of boundary control for thermal convection equations”, Siberian Math. J., 39:5 (1998), 844–858 | DOI | MR | Zbl

[5] Fursikov A. V., Emanuilov Yu. S., “Exact controllability of the Navier–Stokes and Boussinesq equations”, Russian Math. Surveys, 54:3 (1999), 565–618 | DOI | DOI | MR | Zbl

[6] Lee H.-C., Imanuvilov O. Yu., “Analysis of optimal control problems for the 2-D stationary Boussinesq equations”, J. Math. Anal. Appl., 242 (2000), 191–211 | DOI | Zbl

[7] Alekseev G. V., “Solvability of inverse extremal problems for stationary heat and mass transfer equations”, Siberian Math. J., 42:5 (2001), 811–827 | DOI | MR | Zbl

[8] Alekseev G. V., Tereshko D. A., “Stability of optimal controls for the stationary Boussinesq equations”, Inter. J. Differ. Equ., 2011 (2011), 535736 | DOI | Zbl

[9] Abidi H., Zhang P., “On the global well-posedness of 2-D Boussinesq system with variable viscosity”, Adv. Math., 305 (2017), 1202–1249 | DOI | Zbl

[10] Yu Y., Wu X., Tang Y., “Global well-posedness for the 2D Boussinesq system with variable viscosity and damping”, Math. Meth. Appl. Sci., 41:8 (2018), 3044–3061 | DOI | Zbl

[11] Li Z., “Global well-posedness of the 2D Euler-Boussinesq system with stratification effects”, Math. Meth. Appl. Sci., 40:14 (2017), 5212–5221 | DOI | Zbl

[12] Vlasova S. S., Prosviryakov E. Yu., “Two-dimensional convection of an incompressible viscous fluid with the heat exchange on the free border”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:3 (2016), 567–577 | DOI | Zbl

[13] Privalova V. V., Prosviryakov E. Yu., “Couette–Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid, with allowance made for heat recovery”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 22:3 (2018), 532–548 | DOI | Zbl

[14] Burmasheva N. V., Prosviryakov E. Yu., “A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:1 (2017), 180–196 (In Russian) | DOI

[15] Burmasheva N. V., Prosviryakov E. Yu., “A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:4 (2017), 736–751 (In Russian) | DOI | Zbl

[16] Ragulin V. V., “To the problem of viscous fluid flow through bounded region at given pressure of head drop”, Dinamika sploshnoy sredy, 27 (1976), 78–92 (In Russian)

[17] Conca C., Murat F., Pironneau O., “The Stokes and Navier–Stokes equations with boundary conditions involving the pressure”, Japan. J. Math., 20 (1994), 279–318 | DOI | Zbl

[18] Marušić S., “On the Navier–Stokes system with pressure boundary condition”, Ann. Univ. Ferrara, 53 (2007), 319–331 | DOI | Zbl

[19] Bertoluzza S., Chabannes V., Prud'homme C., Szopos M., “Boundary conditions involving pressure for the Stokes problem and applications in computational hemodynamics”, Comput. Methods Appl. Mech. Eng., 322 (2017), 58–80 | DOI

[20] Nečas J., Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg, 2012, xvi+372 pp. | DOI | Zbl

[21] Ladyzhenskaya O. A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, London, Paris, 1969, xviii+224 pp. | Zbl

[22] Renardy M., Rogers R., An Introduction to Partial Differential Equations, 2nd edition, Springer-Verlag, New York, 2004, xiv+434 pp. | DOI

[23] Skrypnik I. V., Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Amer. Math. Soc., Providence, RI, 1994, xii+348 pp. | Zbl