Document Zbl 1185.35337

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V.</a>; <a href="/authors/fedorov.v-e" title="Author Profile">Fedorov, V. E.</a></div> <h2 class="title"> <strong>On the well-posedness of the prediction-control problem for certain systems of equations.</strong> <i>(English. Russian original)</i> <a class="label nowrap" href="/1185.35337">Zbl 1185.35337</a> </h2> <div class="source"> <a href="/serials/157" title="Journal Profile">Math. Notes</a> <a href="/?q=in%3A249848" title="Articles in this Issue">85, No. 3, 426-436 (2009)</a>; translation from Mat. Zametki 85, No. 3, 440-450 (2009). </div> <div class="abstract"><div class="pre">The present paper is devoted to the study of the prediction-observation problem within the framework of an abstract problem for some systems of partial differential equations not resolved for the derivative with respect to time. In particular, the authors investigate inverse problems for a linearized system of Navier-Stokes equations with some overdetermination condition of integral type as well as for a linearized system of equations of the phase field. The authors obtain conditions for the well-defined solvability of these systems. For this purpose they apply many tools from functional analysis, especially from operator theory.</div><div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10745">Jürgen Socolowsky (Brandenburg an der Havel)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5628171">Cited in <strong>12</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R30" title="MSC2020">35R30</a> </td> <td class="space"> Inverse problems for PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76D05" title="MSC2020">76D05</a> </td> <td class="space"> Navier-Stokes equations for incompressible viscous fluids </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q30" title="MSC2020">35Q30</a> </td> <td class="space"> Navier-Stokes equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47N20" title="MSC2020">47N20</a> </td> <td class="space"> Applications of operator theory to differential and integral equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aprediction-control+problem">prediction-control problem</a>; <a href="/?q=ut%3ANavier-Stokes+system+of+equations">Navier-Stokes system of equations</a>; <a href="/?q=ut%3ABanach+space">Banach space</a>; <a href="/?q=ut%3Astrongly+%5C%28%28L%2C+p%29%5C%29-sectorial+operator">strongly $(L, p)$-sectorial operator</a>; <a href="/?q=ut%3Aanalytic+semigroup">analytic semigroup</a>; <a href="/?q=ut%3Aseepage+of+liquids">seepage of liquids</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1185.35337" data-ciurl="/ci/05628171" data-biburl="/bibtex/05628171.bib" data-amsurl="/amsrefs/05628171.bib" data-xmlurl="/xml/05628171.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05628171.pdf" title="Zbl 1185.35337 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1134/S0001434609030134" aria-label="DOI for “On the well-posedness of the prediction-control problem for certain systems of equations”" title="10.1134/S0001434609030134">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">A. N. Tikhonov and V. Y. Arsenin, Methods for the Solution of Ill-Posed Problems (Nauka, Moscow, 1979) [in Russian]. · <a href="/0499.65030" class="nowrap">Zbl 0499.65030</a></td> </tr><tr> <td>[2]</td> <td class="space">M. M. Lavrent’ev, V.G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka Moscow, 1980) [in Russian].</td> </tr><tr> <td>[3]</td> <td class="space">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 href="/1156.35304" class="nowrap">Zbl 1156.35304</a></td> </tr><tr> <td>[4]</td> <td class="space">A. I. Kozhanov, Composite Type Equations and Inverse Problems, in Inverse Ill-posed Probl. Ser. (VSP, Utrecht, 1999).</td> </tr><tr> <td>[5]</td> <td class="space">N. L. Abasheeva, ”Some inverse problems for parabolic equations with changing time direction,” J. Inverse Ill-Posed Probl. 12(4), 337–348 (2004). · <a href="/1081.35132" class="nowrap">Zbl 1081.35132</a> · <a href="https://doi.org/10.1515/1569394042248265" class="nowrap">doi:10.1515/1569394042248265</a></td> </tr><tr> <td>[6]</td> <td class="space">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). · <a href="/1081.35138" class="nowrap">Zbl 1081.35138</a> · <a href="https://doi.org/10.1515/1569394042248210" class="nowrap">doi:10.1515/1569394042248210</a></td> </tr><tr> <td>[7]</td> <td class="space">G. A. Sviridyuk, ”On the general theory of operator semigroups,” UspekhiMat. Nauk 49(4), 47–74 (1994) [RussianMath. Surveys 49 (4), 45–74 (1994)]. · <a href="/0882.47019" class="nowrap">Zbl 0882.47019</a></td> </tr><tr> <td>[8]</td> <td class="space">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)]. · <a href="/0908.47036" class="nowrap">Zbl 0908.47036</a></td> </tr><tr> <td>[9]</td> <td class="space">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). · <a href="/1102.47061" class="nowrap">Zbl 1102.47061</a></td> </tr><tr> <td>[10]</td> <td class="space">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)]. · <a href="https://doi.org/10.4213/sm841" class="nowrap">doi:10.4213/sm841</a></td> </tr><tr> <td>[11]</td> <td class="space">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]. · <a href="/1081.35138" class="nowrap">Zbl 1081.35138</a></td> </tr><tr> <td>[12]</td> <td class="space">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)]. · <a href="/0104.21702" class="nowrap">Zbl 0104.21702</a></td> </tr><tr> <td>[13]</td> <td class="space">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)]. · <a href="/0265.76112" class="nowrap">Zbl 0265.76112</a></td> </tr><tr> <td>[14]</td> <td class="space">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 .</td> </tr><tr> <td>[15]</td> <td class="space">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 href="/0689.35073" class="nowrap">Zbl 0689.35073</a></td> </tr><tr> <td>[16]</td> <td class="space">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)]. · <a href="/0763.35108" class="nowrap">Zbl 0763.35108</a></td> </tr><tr> <td>[17]</td> <td class="space">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)].</td> </tr><tr> <td>[18]</td> <td class="space">S. L. Sobolev, ”On a new problem of mathematical physics,” Izv. Akad. Nauk SSSR Ser. Mat. 18(1), 3–50 (1954). · <a href="/0055.08401" class="nowrap">Zbl 0055.08401</a></td> </tr><tr> <td>[19]</td> <td class="space">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)]. · <a href="/0972.47021" class="nowrap">Zbl 0972.47021</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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