Geodesic
Inverse problems for some Sobolev-type mathematical models
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 9 (2016) no. 2, pp. 75-89 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The present article is devoted to the study of mathematical models based the Sobolev-type equations and systems arising in dynamics of a stratified fluid, elasticity theory, hydrodynamics, electrodynamics, etc. Along with a solution we determine an unknown right-hand side and coefficients in a Sobolev-type equations of the forth order. The overdetermination conditions are the values of a solution in a collection of points of a spatial domain. The problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. The existence and uniqueness theorems of solutions for the linear and nonlinear cases are proven. In the linear case the result is global in time and it is local in the nonlinear case. The main spaces in question are the Sobolev spaces.
Keywords: the Sobolev-type model; Sobolev equation; existence and uniqueness theorems; inverse problem; boundary value problem; plasma waves; rotating fluid; the Boussinesq–Love model. 
@article{VYURU_2016_9_2_a6,
     author = {S. G. Pyatkov and S. N. Shergin},
     title = {Inverse problems for some {Sobolev-type} mathematical models},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {75--89},
     year = {2016},
     volume = {9},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2016_9_2_a6/}
}
TY  - JOUR
AU  - S. G. Pyatkov
AU  - S. N. Shergin
TI  - Inverse problems for some Sobolev-type mathematical models
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2016
SP  - 75
EP  - 89
VL  - 9
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VYURU_2016_9_2_a6/
LA  - en
ID  - VYURU_2016_9_2_a6
ER  - 
%0 Journal Article
%A S. G. Pyatkov
%A S. N. Shergin
%T Inverse problems for some Sobolev-type mathematical models
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2016
%P 75-89
%V 9
%N 2
%U http://geodesic.mathdoc.fr/item/VYURU_2016_9_2_a6/
%G en
%F VYURU_2016_9_2_a6
S. G. Pyatkov; S. N. Shergin. Inverse problems for some Sobolev-type mathematical models. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 9 (2016) no. 2, pp. 75-89. http://geodesic.mathdoc.fr/item/VYURU_2016_9_2_a6/

[1] Sobolev S.L., “On a New Problem of Mathematical Physics”, Izvestiya: Mathematics, 18 (1954), 3–50 (in Russian) | MR | Zbl

[2] Boussinesq J.V., Essai sur la theorie des eaux courantes, Mem. Pesentes Divers Savants Acad. Sci. Inst. France, 23, 1877, 680 pp.

[3] Love A.E.H., A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944 | MR | Zbl

[4] Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner U.D., Linear and Non-Linear Equations Sobolev's Type, Fizmatlit, M., 2007, 736 pp. (in Russian)

[5] Ikezi H., “Experimental Study of Solitons in Plasma”, Solitons in Action, Academic Press, N.Y., 1978, 163–184 [Х. Икези, “Экспериментальное исследование солитонов в плазме”, Солитоны в действии, Мир, М., 1981, 163–184]

[6] Lyubanova A.Sh., “Identification of a Coefficient in the Leading Term of a Pseudoparabolic Equation of Filtration”, Journal of Applied and Industrial Mathematics, 54:6 (2013), 1046–1058 | DOI | MR | Zbl

[7] Lyubanova A.Sh., Tani A., “On Inverse Problems for Pseudoparabolic and Parabolic Equations of Filtration”, Inverse Problems in Science and Engineering, 19:7 (2011), 1023–1042 | DOI | MR | Zbl

[8] Kozhanov A.I., “On Solvability of Inverse Problems of Recovering the Coefficients in Composite Type Equations”, Vestnik Novosibirskogo gosudarstvennogo universiteta. Seriya Matematika, Mekhanika, Informatika, 8:3 (2008), 81–99 (in Russian) | MR | Zbl

[9] Kozhanov A.I., “Nonlinear Loaded Equations and Inverse Problems”, Computational Mathematics and Mathematical Physics, 44:4 (2004), 657–675 | MR

[10] Kozhanov A.I., “On Solvability of Inverse Coefficient for Some Sobolev-Type Equations”, Belgorod State University Scientific Bulletin. Mathematics. Physics, 18:5 (2010), 88–98 | MR

[11] Fedorov V.E., Urazaeva A.V., “An Inverse Problem for Linear Sobolev Type Equations”, Journal of Inverse and Ill-Posed Problems, 12:4 (2004), 387–395 | DOI | MR | Zbl

[12] Ablabekov B.S., Inverse Problems for Pseudoparabolic Equations, Ilim, Bishkek, 2001 (in Russian)

[13] Asanov A., Atamanov E.R., “An Inverse Problem for a Pseudoparabolic Integro-Defferential Operator Equation”, Siberian Mathematical Journal, 38:4 (1995), 645–655 | DOI | MR

[14] Favini A, Lorenzi A., Differential Equations: Inverse and Direct Problems, Chapman and Hall/CRC, Boca Raton–London–N.Y., 2006, 304 pp. | DOI | MR

[15] Zamyshlyaeva A.A., Tsyplenkova O.N., “The Optimal Control over Solutions of the Initial-Finish Value Problem for the Boussinesque–Love Equation”, The Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 5(264), 13–24 (in Russian) | Zbl

[16] Zamyshlyaeva A.A., Yuzeeva A.V., “The Initial-Finish Value Problem for the Boussinesq–Love Equation”, The Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16(192), 23–31 (in Russian) | Zbl

[17] Zamyshlyaeva A.A., Tsyplenkova O.N., “Optimal Control of Solutions of the Showalter–Sidorov–Dirichlet Problem for the Boussinesq–Love Equation”, Differential Equations, 49:11 (2013), 1356–1365 | DOI | MR | Zbl

[18] Mehraliyev Ya.T., “Inverse Problem of the Boussinesq–Love Equation with an Extra Integral Condition”, Journal of Applied and Industrial Mathematics, 16:1(53) (2013), 75–83 | MR

[19] Mehraliyev Ya.T., “On Solvability of an Inverse Boundary Value Problem for the Boussinesq – Love Equation”, Journal of Siberian Federal University. Mathematics and Physics, 6:4 (2013), 485–494 | MR

[20] Ablabekov B.S., Kasymalieva A.A., “An Inverse Problem of Recovering the Right-Hand Side for the Boussinesq–Love Equation”, Theory and Numerical Methods of Solving Inverse and Ill-Posed Problems, Abstracts of the Second International Scientific Conference (September 21–29, 2010), Sobolev Institute of Mathematics, Novosibirsk, 2010, 2–4 (in Russian)

[21] Ladyzhenskaya O.A, Ural'tseva N.N., Linear and Quasilinear Elliptic Equations, Nauka, M., 1964 (in Russian) [О.А. Ладыженская, Н.Н. Уральцева, Линейные и квазилинейные уравнения эллиптического типа, Наука, М., 1973 ] | MR

[22] Gilbarg D., Trudinger N., Elliptic Differential Equation with Partial Derivative of the Second Order, Springer-Verlag, Berlin–Heidelberg, 2001 ; D. Gilbarg, N. Trudinger, Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | MR

[23] Pyatkov S.G., Shergin S.N., “On Some Mathematical Models of Filtration Theory”, The Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 8:2 (2015), 105–116 | MR

[24] Amann H., “Compact Embeddings of Vector-Valued Sobolev and Besov Spaces”, Glasnik matematicki, 35:1 (2000), 161–177 | MR | Zbl

[25] Amann H., “Operator-Valued Foutier Multipliers, Vector-Valued Besov Spaces and Applications”, Mathematische Nachrichten, 186:1 (1997), 5–56 | DOI | MR | Zbl