Some Inverse Problems for Convection-Diffusion Equations

We examine the well-posedness questions for some inverse problems in the mathematical models of heat-and-mass transfer and convection-diffusion processes. The coefficients and right-hand side of the system are recovered under certain additional overdetermination conditions, which are the integrals of a solution with weights over some collection of domains. We prove an existence and uniqueness theorem, as well as stability estimates. The results are local in time. The main functional spaces used are Sobolev spaces. These results serve as the base for justifying of the convergence of numerical algorithms for inverse problems with pointwise overdetermination, which arise, in particular, in the heat-and-mass transfer problems on determining the source function or the parameters of a medium.

Keywords

parabolic system; convection-diffusion; heat-and-mass transfer; inverse problem; control problem; boundary value problem; well-posedness.

References

1. Alekseev G.V. Optimizatsiya v statsionarnykh zadachakh teplomassoperenosa i magnitnoy gidrodinamiki [Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics]. Moscow, Nauchnyi Mir, 2010.
2. Belov Ya.Ya. Inverse Problems for Parabolic Equations. Utrecht, VSP, 2002.
3. Babeshko O.M., Evdokimova O.V., Evdokimov S.M. On Taking into Account the Types of Sources and Settling Zones of Pollutants. Doklady Mathematics, 2000, vol. 61, no. 2, pp. 283-285.
4. Kalinina E.A. A Numerical Study of an Inverse Problem of Recovering the Heat Source for the Two-Dimensional Nonstationary Convection-Diffusion Equation. Far Eastern Mathematical Journal, 2004, vol. 5, no. 1, pp. 89-99. (in Russian)
5. Kriksin Yu.A., Plushchev S.N., Samarskaya E.A., Toshkin V.F. Inverse Problem of Recovering the Source Density for a Convection-Diffusion Equation. Matematicheskoe Modelirovanie [Mathematical Modelling], 1995, vol. 7, no. 11, pp. 95-108. (in Russian)
6. Iskenderov A.D., Akhundov A.Ya. Inverse Problem for a Linear System of Parabolic Equations. Doklady Mathematics, 2009, vol. 79, no. 1, pp. 73-75. DOI: 10.1134/S1064562409010219
7. Ismailov M.I., Kanca F. Inverse Problem of Finding the Time-Dependent Coefficient of Heat Equation from Integral Overdetermination Condition Data. Inverse Problems In Science and Engineering, 2012, vol. 20, no. 24, pp. 463-476.
8. Ivanchov M.I. Inverse Problem of Simulataneous Determination of Two Coefficients in a Parabolic Equation. Ukrainian Mathematical Journal, 2000, vol. 52, no. 3, pp. 379-387. DOI: 10.1007/BF02513132
9. Jing Li, Youjun Xu. An Inverse Coefficient Problem with Nonlinear Parabolic Equation. Journal of Applied Mathematics and Computing, 2010, vol. 34, no. 1-2, pp. 195-206. DOI: 10.1007/s12190-009-0316-8
10. Kamynin V.L., Franchini E. An Inverse Problem for a Higher-Order Parabolic Equation. Mathematical Notes, 1998, vol. 64, no. 5, pp. 590-599. DOI: 10.1007/BF02316283
11. Kerimov N.B., Ismailov M.I. An Inverse Coefficient Problem for the Heat Equation in the Case of Nonlocal Boundary Conditions. Journal of Mathematical Analysis and Applications, 2012, vol. 396, issue 2, pp. 546-554. DOI: 10.1016/j.jmaa.2012.06.046
12. Kozhanov A.I. Parabolic Equations with an Unknown Coeffiecient Depending on Time. Computational Mathematics and Mathematical Physics, 2005, vol. 45, no. 12, pp. 2085-2101.
13. Vasin I.A., Kamynin V.L. On the Asymptotic Behavior of Solutions to Inverse Problems for Parabolic Equations. Siberian Mathematical Journal, 1997, vol. 38, no. 4, pp. 647-662. DOI: 10.1007/BF02674572
14. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for solving inverse problems in Mathematical Physics. New-York, Marcel Dekker, Inc. 1999.
15. Tryanin A.P. Determination of Heat-Transfer Coefficients at the Inlet into a Porous Body and Inside It by Solving the Inverse Problem. Journal of Engineering Physics, 1987, vol. 52, no. 3, pp. 346-351. DOI: 10.1007/BF00872521
16. Dehghan M., Shakeri F. Method of Lines Solutions of the Parabolic Inverse Problem with an Overspecification at a Points. Numerical Algorithms, 2009, vol. 50, no. 4, pp. 417-437. DOI: 10.1007/s11075-008-9234-3
17. Pyatkov S.G., Samkov M.L. On Some Classes of Coefficient Inverse Problems for Parabolic Systems of Equations. Sib. Adv. in Math., 2012, vol. 22, no. 4, pp. 287-302. DOI: 10.3103/S1055134412040050
18. Pyatkov S.G. On Some Classes of Inverse Problems for Parabolic Equations. Journal of Inverse Ill-Posed problems, 2011, vol. 18, no. 8, pp. 917-934. DOI: 10.1515/jiip.2011.011
19. Ivanchov M. Inverse Problems for Equations of Parabolic Type Math. Studies. Monograph Series. V. 10. Lviv, WNTL Publishers, 2003.
20. Kabanikhin S.I. Obratnye i nekorektnye zadachi [Inverse and Illl-posed problems]. Novosibirsk, Sibirskoe Nauchn. Izdat., 2009. 457 p.
21. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, 18. Amsterdam, North-Holland Publishing, 1978.
22. Amann H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces. Glasnik matematicki, 2000. vol. 35, no. 55, pp. 161-177.
23. Amann H. Operator-Valued Foutier Multipliers, Vector-Valued Besov Spaces and Applications. Mathematische Nachrichten, 1997, vol. 186, no. 1, pp. 5-56. DOI: 10.1002/mana.3211860102
24. Amann H. Linear and Quasilinear Parabolic Problems. V. I. Basel, Boston, Berlin, Birkhauser Verlag, 1995. DOI: 10.1007/978-3-0348-9221-6
25. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Lineynye i kvazilineynye uravneniya parabolicheskogo tipa [Linear and Quasilinear Equations of the Parabolic Type]. Moscow, Nauka, 1967.
26. Ladyzhenskaya O.A., Ural'tseva N.N. Lineynye i kvazilineynye uravneniya ellipticheskogo tipa [Linear and Quasilinear Elliptic Equation of the Elliptic Type]. Moscow, Nauka, 1973.
27. Alifanov O.M., Artyukhov E.A., Nenarokom A.V. Obratnye zadachi slozhnogo teploobmena [Inverse Problems of Complicated Heat Exchange]. Moscow, Yanus-K, 2009.
28. Alifanov O.M. Inverse Heat Transfer Problems. Springer-Verlag, Berlin Heidelberg, 1994. DOI: 10.1007/978-3-642-76436-3
29. Ozisik M.N., Orlando H.A.B. Inverse Heat Transfer. New-York, Taylor & Francis, 2000.