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On the regular solvability of some classes of transmission problems in a cylindrical space domain. (English) Zbl 1461.35121

Summary: In the article we examine the questions of regular solvability in the Sobolev spaces of the transmission problems with transmission conditions of imperfect contact type for parabolic second order systems in cylindrical space domains. A solution has all generalized derivatives occurring in the system summable to some power \(p \in(1,\infty )\). At the interface the limit values of the conormal derivatives are expressed through the limit values of a solution. The problem does not belong to the class of classical diffraction problems and arises when describing heat-and-mass transfer processes in layered media. The proof relies on a priori bounds and the method of continuation in a parameter.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35R05 PDEs with low regular coefficients and/or low regular data
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[1] H.D. Baehr, K. Stephan, Heat and mass transfer, Springer, Berlin, 1998. Zbl 0948.80001 · Zbl 0948.80001
[2] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, 23, AMS, Providence, 1968. Zbl 0174.15403 · Zbl 0174.15403
[3] O.A. Ladyzhenskaya, V.Ya. Rivkind, N.N. Ural’tseva, Classical solvability of diraction problems in the case of elliptic and parabolic equations, Sov. Math., Dokl., 5 (1965), 12491252. Zbl 0163.13402
[4] O.A. Ladyzhenskaya, V.Ya. Rivkind, N.N. Ural’tseva, The classical solvability of diraction problems, Proc. Mat. Inst. Steklova 92 (1968), 132166. Zbl 0165.11802
[5] O.A. Ladyzhenskaya, On non-stationary operator equations and their applications to linear problems of mathematical physics, Mat. Sb., N.Ser., 45(87) (1958), 123158. Zbl 0081.09702
[6] O.A. Oleinik, Equations of elliptic and parabolic type with discontinuous coecients, UMN, 14:5(89) (1959), 164166.
[7] O.A. Oleinik, Boundary-value problems for linear elliptic and parabolic equations with discontinuous coecients, Transl., Ser. 2, Am. Math. Soc., 42 (1964), 175194. Zbl 0148.35302
[8] Z.G. Sheftel’, Solvability inLpand classical solvability of general boundary-value problems for elliptic equations with discontinuous coecients, UMN, 19:4(118) (1964), 230232.
[9] Z.G. Sheftel’, Estimates inLpof solutions of elliptic equations with discontinuous coecients and satisfying general boundary conditions and conjugacy conditions, Sov. Math., Dokl., 4 (1963), 321324. Zbl 0163.34901
[10] M. Schechter, A generalization of the problem of transmission, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser., 14 (1960), 207236. Zbl 0094.29603 · Zbl 0094.29603
[11] N.V. Zhitarashu, Apriori estimates and solvability of general boundary value problems for general elliptic systems with discontinuous coecients, Sov. Math., Dokl., 6 (1965), 13941397. Zbl 0144.14802 · Zbl 0144.14802
[12] N.V. Zhitarashu, Schauder estimates and solvability of general boundary problems for general parabolic systems with discontinuous coecients, Sov. Math., Dokl., 7 (1966), 952956 Zbl 0168.08303 · Zbl 0168.08303
[13] J. Pruss, G. Simonett, Moving interfaces and quasilinear parabolic evolution equations, Monographs in Mathematics, 105, Birkhauser Publishing, Basel, 2016. Zbl 1435.35004 · Zbl 1435.35004
[14] L. Simon On contact problems for nonlinear parabolic functional dierential equations, Electron. J. Qual. Theory Dier. Equ., 22 (2003). Zbl 1068.35177
[15] V.A. Belonogov, S.G. Pyatkov, On solvability of conjugation problems with non-ideal contact conditions, Russ. Math., 64:7 (2020), 1326. Zbl 07309104 · Zbl 1459.35235
[16] D.A. Nomirovskii, Generalized solvability of parabolic systems with nonhomogeneous transmission conditions of nonideal contact type, Dier. Equ., 40:10 (2004), 14671477. Zbl 1182.35138 · Zbl 1182.35138
[17] B.S. Jovanovic, L.G. Vulkov, Formulation and analysis of a parabolic transmission problem on disjoint intervals, Publ. Inst. Math., Nouv. Ser., 91 (2012), 111123. Zbl 1299.35149 · Zbl 1299.35149
[18] L.B. Drenchev, J. Sobczak, Determination of the heat exchange coecient on the castingdie interface, High temperature capillarity: reviewed proceedings of the Second International Conference HTC-97 held in Cracow, Poland, 29 June - 2 July, 1997.
[19] M.N. Ozisik, H.R.B. Orlando, Inverse heat transfer, Taylor & Francis, New York, 2000. ON SOLVABILITY OF SOME CLASSES OF TRANSMISSION PROBLEMS205
[20] A. Abreu, H.R.B. Orlande, C.P. Naveira-Cotta, J.N.N. Quaresma, R.M. Cotta, Identication of contact failures in multi-layered composites, Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2011 August 28-31, 19 (2011).
[21] J.Jr. Lugon, A.J.S. Neto, An inverse problem of parameter estimation in simultaneous heat and mass transfer in a one-dimensional porous medium, Proceedings of COBEM 2003. 17th International Congress of Mechanical Engineering, November 10-14, 111 (2003).
[22] R.I. Hickson, S.I. Barry, G.N. Mercer, Critical times in multilayer diusion. I: Exact solutions, Int. J. Heat Mass Transfer, 52:2526 (2009), 57765783. Zbl 1177.80022 · Zbl 1177.80022
[23] C.H. Huang, T.M. Ju, An inverse problem of simultaneously estimating contact conductance and heat transfer coecient of exhaust gases between engine’s exhaust valve and seat, Int. J. Numer. Methods Engineering, 38:5 (1995), 735-754.
[24] N.M. AL-Najem, Whole time domain solution of inverse heat conduction problem in multilayer media, Heat and Mass Transfer, 33:3 (1997), 233-240.
[25] A.M. Osman, J.V. Beckf, Nonlinear inverse problem for the estimation of time-and-spacedependent heat-transfer coecients, J. Thermophysics, 3:2 (2015), 146-152.
[26] D.B. Rodriguesa, P.J.S. Pereira, P. Limro-Vieira, P.R. Stauer, P.F. Maccarini, Study of the one dimensional and transient bioheat transfer equation: Multi-layer solution development and applications, Int. J. Heat Mass Transf., 62 (2013), 153162.
[27] L. Zhuo, D. Lesnik, S. Meng, Reconstruction of the heat transfer coecient at the interface of a bi-material, Inverse Problems in Science and Engineering, 28:3 (2020), 374401. · Zbl 1466.80010
[28] M.S. Agranovich, M.I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russ. Math. Surv., 19:3 (1964), 53161. Zbl 0137.29602 · Zbl 0137.29602
[29] H. Triebel, Interpolation theory, function spaces, dierential operators, North-Holland Mathematical Library, 18, North-Holland Publishing Company, Amsterdam etc., 1978. Zbl 0387.46032 · Zbl 0387.46032
[30] R. Denk, M. Hieber, J. Pruss, OptimalLp−Lq-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257:1 (2007), 193224. Zbl 1210.35066 · Zbl 1210.35066
[31] M.A. Verzhbitskii, S.G. Pyatkov, On some inverse problems of determining boundary regimes, Mat. Zamet. SVFU, 23:2 (2016), 318. Zbl 1399.35368 · Zbl 1399.35368
[32] P. Grisvard, Equations dierentielles abstraites, Ann. Sci. Ec. Norm. Sup., 4:2 (1969), 311 395. MR0270209 · Zbl 0193.43502
[33] V.A. Solonnikov, On boundary value problems for linear parabolic systems of dierential equations of general form, Proc. Steklov Inst. Math., 83 (1965), 3163. Zbl 0164.12502
[34] H. Triebel, Theory of function spaces, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1983. Zbl 0546.46028 · Zbl 0546.46028
[35] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in H.-J. Schmeisser (ed.) et al., Function spaces, dierential operators and nonlinear analysis. Survey articles and communications of the international conference held in Friedrichsroda, Germany, September 20-26, 1992, B. G. Teubner Verlagsgesellschaft. TeubnerTexte Math., 133, Stuttgart, 1993. Zbl 0810.35037 · Zbl 0810.35037
[36] G.M. Lieberman, Second order parabolic dierential equations, World Scientic, Singapure, 1996. Zbl 0884.35001 · Zbl 0884.35001
[37] V.A. Kondrat’ev, O.A. Olejnik, Boundary-value problems for partial dierential equations in non-smooth domains, Russ. Math. Surv., 38:2 (1983), 186. Zbl 0548 · Zbl 0548.35018
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