Abstract
We consider the inverse problem for equations of Sobolev type and their applications to linearized Navier-Stokes systems and phase-field systems. We obtain conditions for the well-defined solvability of these systems.
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A. N. Tikhonov and V. Y. Arsenin, Methods for the Solution of Ill-Posed Problems (Nauka, Moscow, 1979) [in Russian].
M. M. Lavrent’ev, V.G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka Moscow, 1980) [in Russian].
A. I. Prilepko, “The semigroup method for inverse, nonlocal and nonclassical problems: Prediction-control and prediction-observation for evolution equations, I,” Differentsial’nye Uravneniya 41(11), 1560–1571 (2005) [Differential Equations 41 (11), 1635–1646 (2005)].
A. I. Kozhanov, Composite Type Equations and Inverse Problems, in Inverse Ill-posed Probl. Ser. (VSP, Utrecht, 1999).
N. L. Abasheeva, “Some inverse problems for parabolic equations with changing time direction,” J. Inverse Ill-Posed Probl. 12(4), 337–348 (2004).
V. E. Fedorov and A. V. Urazaeva, “An inverse problem for linear Sobolev type equations,” J. Inverse Ill-Posed Probl. 12(4), 387–395 (2004).
G. A. Sviridyuk, “On the general theory of operator semigroups,” UspekhiMat. Nauk 49(4), 47–74 (1994) [RussianMath. Surveys 49 (4), 45–74 (1994)].
G. A. Sviridyuk and V. E. Fedorov, “On the identities of analytic semigroups of operators with kernels,” Sibirsk. Mat. Zh. 39(3), 604–616 (1998) [Siberian Math. J. 39 (3), 522–533 (1998)].
G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, in Inverse Ill-posed Probl. Ser. (VSP, Utrecht, 2003).
V. E. Fedorov, “Holomorphic resolving semigroups of Sobolev-type equations in locally convex spaces,” Mat. Sb. 195(8), 131–160 (2004) [Russian Acad. Sci. Sb. Math. 195 (8), 1205–1234 (2004)].
V. E. Fedorov and A. V. Urazaeva, “An inverse problem for a class of singular linear operator-differential equations,” in Proc. Voronezh Winter Workshop (Voronezh Gos. Univ. Voronezh, 2004), pp. 161–172 [in Russian].
G. I. Barenblatt, Yu. P. Zheltov, and I. N. Kochina, “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata),” Prikl.Mat. Mekh. 24(5), 852–864 (1960) [J. Appl.Math.Mech. 24, 1286–1303 (1961)].
E. S. Dzektser, “Generalization of the equation of motion of ground waters with a free surface,” Dokl. Akad. Nauk SSSR 202(5), 1031–1033 (1972) [SovietMath. Dokl. 17, 108–110 (1972)].
V. E. Fedorov and A. V. Urazaeva, “Inverse problems for certain nonclassical equations of mathematical physics,” in Information Technologies and Inverse Problems of Rational Nature Management Proceedings of the conference “Constructive Methods in the Theory of Inverse Problems” (YuNIIIT, Khanty-Mansiisk, 2005), Vol. 1, pp. 71–73 [in Russian]; http://www.uriit.ru/conf_erohin_50/Part_01_16.pdf.
A. I. Prilepko and I. A. Vasin, “Inverse initial-boundary-value problems for linearized nonstationary Navier-Stokes equations,” Differentsial’nye Uravneniya 25(1), 106–117 (1989) [Differential Equations 25 (1), 85–92 (1989)].
A. I. Prilepko and I. A. Vasin, “Some time-dependent inverse problems of hydrodynamics with final observation,” Dokl. Akad. Nauk SSSR 314(5), 1075–1078 (1990) [Soviet Math. Dokl. 42 (2), 604–607 (1990)].
P. I. Plotnikov and V. N. Starovoitov, “The Stefan problem with surface tension as the limit of a phase field model,” Differentsial’nye Uravneniya 29(3), 461–471 (1993) [Differential Equations 29 (3), 395–404 (1993)].
S. L. Sobolev, “On a new problem of mathematical physics,” Izv. Akad. Nauk SSSR Ser. Mat. 18(1), 3–50 (1954).
G. A. Sviridyuk and G. A. Kuznetsov, “Relatively strongly p-sectorial linear operators,” Dokl. Ross. Akad. Nauk 365(6), 736–738 (1999) [Russian Acad. Sci. Dokl.Math. 59 (2), 298–300 (1999)].
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Original Russian Text © A. V. Urazaeva, V. E. Fedorov, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 3, pp. 440–450.
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Urazaeva, A.V., Fedorov, V.E. On the well-posedness of the prediction-control problem for certain systems of equations. Math Notes 85, 426–436 (2009). https://doi.org/10.1134/S0001434609030134
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DOI: https://doi.org/10.1134/S0001434609030134