On some classes of inverse problems on determining source functions for heat and mass transfer systems

S. G. Pyatkov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Modeling, Tome 188 (2020), pp. 23-42.

Résumé

In this paper, we considers the question on the well-posedness in Sobolev spaces of the problem of determining the source function in a heat and mass transfer system consisting of the Navier–Stokes system, a parabolic equation for temperature, and a parabolic system for the concentrations of substances being transferred. Weighted integrals solution over the spatial domain serve as overdetermination conditions. A local (in time) existence theorem for the solution of the problem in the nonlinear case is proved and stability estimates are obtained; for a linearized system, a global existence theorem is obtained.

Détail

Keywords: heat and mass transfer system, inverse problem, integral overdetermination, uniqueness, initial-boundary-value problem.
Mots-clés : existence

Bibliographie

[1] Alekseev G. V., Optimizatsiya v statsionarnykh zadachakh teplomassoperenosa i magnitnoi gidrodinamiki, Nauchnyi mir, M., 2010

[2] Alifanov O. M., Artyukhin E. A., Nenarokomov A. V., Obratnye zadachi v issledovanie slozhnogo teploobmena, Yanus-K, M., 2009

[3] Iskenderov A. D., Akhundov A. Ya., “Obratnaya zadacha dlya lineinoi sistemy parabolicheskikh uravnenii”, Dokl. RAN., 424:4 (2005), 442–444

[4] Kamynin V. L., Franchini E., “Ob odnoi obratnoi zadache dlya parablolicheskogo uravneniya vysokogo poryadka”, Mat. zametki., 64:5 (1998), 680–691 | MR | Zbl

[5] Kozhanov A. I., “Parabolicheskie uravneniya s neizvestnym koeffitsientom, zavisyaschim ot vremeni”, Zh. vychisl. mat. mat. fiz., 45:12 (2005), 2168–2184 | MR | Zbl

[6] Korotkova E. M., Pyatkov S. G., “O nekotorykh obratnykh zadachakh dlya linearizovannoi sistemy teplomassoperenosa”, Mat. tr., 17:2 (2014), 142–162 | MR | Zbl

[7] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[8] Lykov A. V., Mikhailov Yu. A., Teoriya teplo- i massoperenosa, Gosenergoizdat, Leningrad, 1963

[9] Polezhaev V. I., Bune A. V., Verezub N. A. i dr., Matematicheskoe modelirovanie konvektivnogo teplomassoobmena na osnove uravnenii Nave—Stoksa, Nauka, M., 1987

[10] Prilepko A. I., Ivankov A. L., Solovev V. V., “Obratnye zadachi dlya uravnenii perenosa i parabolicheskikh uravnenii”, Edinstvennost, ustoichivost i metody resheniya nekorrektnykh zadach matematicheskoi fiziki, Vychislitelnyi tsentr SO RAN, Novosibirsk, 1984, 137–142

[11] Pyatkov S. G., Safonov E. I., “O nekotorykh klassakh lineinykh obratnykh zadach dlya parabolicheskikh sistem uravnenii”, Nauch. ved. BelGU., 35:7 (183) (2014), 61–-75

[12] Pyatkov S. G., Safonov E. I., “O nekotorykh klassakh lineinykh obratnykh zadach dlya parabolicheskikh sistem uravnenii”, Sib. elektron. izv., 11 (2014), 777–-779

[13] Pyatkov S. G., “O nekotorykh obratnykh zadachakh dlya operatorno-differentsialnykh uravnenii pervogo poryadka”, Sib. mat. zh., 60:1 (2019), 183–193 | MR | Zbl

[14] Solonnikov V. A., “O kraevykh zadachakh dlya lineinykh parabolicheskikh sistem differentsialnykh uravnenii obschego vida”, Tr. Mat. in-ta im. V. A. Steklova AN SSSR., 83 (1965), 3–-163

[15] Abels H., “Reduced and generalized Stokes resolvent equations in asymptotically flat layers. Part I: Unique solvability”, J. Math. Fluid Mech., 7:2 (2005), 201-–222 | DOI | MR | Zbl

[16] Abels H., “Reduced and generalized Stokes resolvent equations in asymptotically flat layers. Part II: $H^{\infty}$-calculus”, J. Math. Fluid Mech., 7:2 (2005), 223–-260 | DOI | MR | Zbl

[17] Abels H., “Bounded imaginary powers and $H^{\infty}$-calculus of the Stokes operator in unbounded domains”, Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann, Birkhäuser Verlag, Basel, 2005, 1–15 | MR | Zbl

[18] Amann H., Linear and Quasilinear Parabolic Problems, Birkhäuser Verlag, Basel, 2005 | MR

[19] Amann H., “Compact embeddings of vector-valued Sobolev and Besov spaces”, Glas. Mat., 35 (55) (2000), 161-–177 | MR | Zbl

[20] Bejan A., Convection heat transfer, Wiley, New York, 2004

[21] Belov Ya. Ya., Inverse Problems for Parabolic Equations, VSP, Utrecht, 2002 | MR

[22] Cannon J. R., “A class of non-linear nonclassical parabolic equations”, J. Differ. Equ., 79:2 (1989), 266–288 | DOI | MR | Zbl

[23] Capatina A., Stavre R., “A control problem in biconvective flow”, J. Math. Kyoto Univ., 37:4 (1997), 585–595 | MR | Zbl

[24] Loucks D. P., van Beek E. et al., Water Resources Systems: Planning and Management, UNESKO Publishing, Paris–Delft, 2005

[25] Denk R., Hieber M., Prüss J., “$R$-boundedness, Fourier multipliers, and problems of elliptic and parabolic type”, Mem. Am. Math. Soc., 166 (2003), 1–114 | MR

[26] Desch W., Hieber M., and Prüss J., “$L_{p}$-Theory of the Stokes equation in a half space”, J. Evol. Equ., 1 (2001), 115–142 | DOI | MR | Zbl

[27] Denk R., Hieber M., and Prüss J., “Optimal $L_{p}-L_{q}$-estimates for parabolic boundary-value problems with inhomogeneous data”, Math. Z., 257:1 (2007), 193–224 | DOI | MR | Zbl

[28] Ewing R. E., Lin T., “A class of parameter estimation techniques for fluid flow in porous media”, Adv. Water Res., 14:2 (1991), 89–97 | DOI | MR

[29] Farwig R., Myong-Hwan R., “Resolvent estimates and maximal regularity in weighted $L_{q}$-spaces of the Stokes operator in an infinite cylinder”, J. Math. Fluid Mech., 10:3 (2008), 352–387 | DOI | MR | Zbl

[30] Farwig R., Myong-Hwan R., “The resolvent problem and $H^{\infty}$-calculus of the Stokes operator in unbounded cylinders with several exits to infinity”, J. Evol. Equ., 7:3 (2007), 497–528 | DOI | MR | Zbl

[31] Frölich A., “The Stokes operator in weighted $L_q$-spaces, II: Weighted resolvent estimates and maximal $L_{p}$-regularity”, Math. Ann., 339:2 (2007), 287–316 | DOI | MR

[32] Giga Y., Sohr H., “On the Stokes operator in exterior domains”, J. Fac. Sci. Univ. Tokyo. Sect. IA. Math., 36:2 (1989), 103–130 | MR

[33] Giga Y., Sohr H., “Abstract $L_{p}$-estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains”, J. Funct. Anal., 102:1 (1991), 72–94 | DOI | MR | Zbl

[34] Grisvard P., “Equations operationnelles dans les de Banach et preblemes aux limites dans des ouverts cylindriques”, Ann. Scu. Norm. Super. Pisa., 21:3 (1967), 308–347 | MR

[35] Hazanee A., Lesnic D., Ismailov M. I., Kerimov N. B., “Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions”, Appl. Math. Comput., 346 (2019), 800–815 | MR | Zbl

[36] Hasanov A., Hasanoglu S., “Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry”, J. Math. Chem., 44:3 (2008), 731–742 | DOI | MR | Zbl

[37] Hussein M. S., Lesnic D., “Simultaneous determination of time-dependent coefficients and heat source”, Int. J. Comput. Meth. Eng. Sci. Mech., 17:5–6 (2016), 401–411 | DOI | MR

[38] Joseph D. D., Stability of Fluid Motions, Springer-Verlag, Berlin–Heidelberg–New York, 1976 | MR

[39] Isakov V., Inverse problems for partial differential equations, Springer-Verlag, Berlin–Heidelberg–New York, 2006 | MR | Zbl

[40] Ismailov M. I., Kanca F., “Inverse problem of finding the time-dependent coefficient of heat equation from integral overdetermination condition data”, Inv. Probl. Sci. Eng., 20:4 (2012), 436–476 | MR

[41] Ismailov M., Erkovan S., “Inverse problem of finding the coefficient of the lowest term in two-dimensional heat equation with Ionkin-type boundary condition”, Zh. vychisl. mat. mat. fiz., 59:5 (2019), 859 | MR

[42] Isakov V., Inverse problems for equations of parabolic type, Springer, Berlin, 2003 | MR

[43] Jing Li, Youjun Xu, “An inverse coefficient problem with nonlinear parabolic equation”, J. Appl. Math. Comput., 34:1–2 (2010), 195–206 | DOI | MR | Zbl

[44] Joseph D. D., Stability of fluid motions, II, Springer-Verlag, Berlin–Heidelberg–New York, 1976 | MR | Zbl

[45] Kabanikhin S. I., Inverse and Ill-Posed Problems, de Gruyter, Berlin–Boston, 2012 | MR | Zbl

[46] Kerimov N. B., Ismailov M. I., “An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions”, J. Math. Anal. Appl., 396:2 (2012), 546–554 | DOI | MR | Zbl

[47] Kozhanov A. I., Composite type equations and inverse problems, VSP, Utrecht, 1999 | MR | Zbl

[48] Levandowsky M., Childress W. S., Hunter S. H., Spiegel E. A., “A mathematical model of pattern formation by swimming microorganisms”, J. Protozoology., 22:2 (1975), 296–306 | DOI

[49] Ozisik M. N., Orlando H. A. B., Inverse Heat Transfer, Taylor Francis, New York, 2000 | MR

[50] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 1999 | MR

[51] Pyatkov S. G., Safonov E. I., “Some inverse problems for convection-diffusion systems of equations”, Vestn. YuUrGU. Ser. mat. model. programm., 7:4 (2014), 36–50 | MR | Zbl

[52] Pyatkov S. G., Safonov E. I., “Solvability of some inverse problems for the nonstationary heat-and-mass-transfer system”, J. Math. Anal. Appl., 446:2 (2017), 1449–1465 | DOI | MR | Zbl

[53] Solonnikov V. A., “Estimates of solutions of the Stokes equations in Sobolev spaces with a mixed norm”, J. Math. Sci., 123:6 (2004), 4637–4653 | DOI | MR

[54] Triebel H., Interpolation Theory. Function Spaces. Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978 | MR

[55] Triebel H., Theory of Function Spaces, Birkhäuser-Verlag, Basel, 1983 | MR | Zbl