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Mat. Sb. (N.S.), 1981, Volume 115(157), Number 2(6), Pages 281–306 (Mi msb2385)  

This article is cited in 57 scientific papers (total in 57 papers)

Control problems and theorems concerning the unique solvability of a mixed boundary value problem for the three-dimensional Navier–Stokes and Euler equations

A. V. Fursikov


Abstract: In this paper, theorems on the existence of smooth solutions of certain control problems describable by the Navier–Stokes and Euler equations are proved. It is shown that a mixed boundary value problem for the Navier–Stokes and Euler equations of dimension $n\geqslant3$ is uniquely solvable for a dense set of right-hand sides.
Bibliography: 13 titles.

Full text: PDF file (2389 kB)
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English version:
Mathematics of the USSR-Sbornik, 1982, 43:2, 251–273

Bibliographic databases:

UDC: 517.946
MSC: Primary 35B37, 35G30; Secondary 35Q10
Received: 13.06.1980

Citation: A. V. Fursikov, “Control problems and theorems concerning the unique solvability of a mixed boundary value problem for the three-dimensional Navier–Stokes and Euler equations”, Mat. Sb. (N.S.), 115(157):2(6) (1981), 281–306; Math. USSR-Sb., 43:2 (1982), 251–273

Citation in format AMSBIB
\Bibitem{Fur81}
\by A.~V.~Fursikov
\paper Control problems and theorems concerning the unique solvability of a~mixed boundary value problem for the three-dimensional Navier--Stokes and Euler equations
\jour Mat. Sb. (N.S.)
\yr 1981
\vol 115(157)
\issue 2(6)
\pages 281--306
\mathnet{http://mi.mathnet.ru/msb2385}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=622148}
\zmath{https://zbmath.org/?q=an:0491.49006|0478.49010}
\transl
\jour Math. USSR-Sb.
\yr 1982
\vol 43
\issue 2
\pages 251--273
\crossref{https://doi.org/10.1070/SM1982v043n02ABEH002447}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Fursikov, “Properties of solutions of some extremal problems connected with the Navier–Stokes system”, Math. USSR-Sb., 46:3 (1983), 323–351  mathnet  crossref  mathscinet  zmath
    2. Fursikov A., “Properties of Solutions of Some Control-Problems Connected with the Navier–Stokes System”, 262, no. 1, 1982, 44–48  mathscinet  isi
    3. Fursikov A., “Space-Time Moments and Statistical Solutions on Smooth Solutions of the 3-Dimension Navier–Stokes System Or a Quasi-Linear Parabolic-System”, 274, no. 3, 1984, 548–553  mathscinet  zmath  isi
    4. J. Lions, “Some questions in the optimal control of distributed systems”, Russian Math. Surveys, 40:4 (1985), 57–72  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. Alexandrov Y., “The Existence of Solution of a Certain Class of Control-Problems for Distributed Parameter-Systems”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1985, no. 5, 24–28  isi
    6. Lions J., “Distributed Systems with Uncomplete Data and Lagrange Multipliers”, Lect. Notes Math., 1190 (1986), 273–305  crossref  mathscinet  zmath  isi
    7. Lions J., “Some Remarks on the Optimal-Control of Singular Distributed Systems”, 45, no. Part 2, 1986, 127–153  mathscinet  zmath  isi
    8. A. V. Fursikov, “On uniqueness of the solution of the chain of moment equations corresponding to the three-dimensional Navier–Stokes system”, Math. USSR-Sb., 62:2 (1989), 465–490  mathnet  crossref  mathscinet  zmath
    9. Bilic N., “Necessary Conditions for an Optimal Distributed Control Problem”, Boll. Unione Mat. Italiana, 1A:2 (1987), 247–254  mathscinet  isi
    10. Romanovich A., Fursikov A., “On Cauchy-Problem Solvability for the Laplace Operator”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1987, no. 3, 78–80  mathscinet  zmath  isi
    11. Ivanenko V., Melnik V., “On the Optimal-Control of Evolutional Equations with Multivalued Operators”, 297, no. 1, 1987, 26–30  mathscinet  isi
    12. Ivanenko V., Melnik V., “Extremal Solutions of the Operator-Differential Inclusions”, no. 5, 1987, 72–75  isi
    13. Emanuilov O., “On Solution Uniqueness Question for Control-Problems”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1989, no. 1, 85–88  mathscinet  zmath  isi
    14. Yu. S. Èmanuilov, “Existence theorems for control problems of singular systems with distributed parameters”, Math. USSR-Sb., 69:2 (1991), 341–355  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    15. Melnik V., “On Extremal Problems for Differential-Operator Systems”, 314, no. 1, 1990, 101–105  mathscinet  isi
    16. Fattorini H., Sritharan S., “Existence of Optimal Controls for Viscous-Flow Problems”, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 439:1905 (1992), 81–102  crossref  mathscinet  zmath  adsnasa  isi
    17. Fursikov A., “Lagrange Principle for Problems of Optimal-Control of Ill-Posed Or Singular Distributed Systems”, J. Math. Pures Appl., 71:2 (1992), 139–195  mathscinet  zmath  isi
    18. L. S. Hou, Y. Yan, “Dynamics for Controlled Navier–Stokes Systems with Distributed Controls”, SIAM J Control Optim, 35:2 (1997), 654  crossref  mathscinet  zmath  isi
    19. L. S. Hou, Y. Yan, “Dynamics and Approximations of a Velocity Tracking Problem for the Navier–Stokes Flows with Piecewise Distributed Controls”, SIAM J Control Optim, 35:6 (1997), 1847  crossref  mathscinet  zmath  isi  elib
    20. L. S. HOU, S. S. RAVINDRAN, Y. YAN, “Numerical Solution of Optimal Distributed Control Problems for Incompressible Flows”, International Journal of Computational Fluid Dynamics, 8:2 (1997), 99  crossref
    21. Björn Schmalfuss, “Qualitative properties for the stochastic Navier–Stokes equation”, Nonlinear Analysis: Theory, Methods & Applications, 28:9 (1997), 1545  crossref
    22. Franco Flandoli, “Irreducibility of the 3-D Stochastic Navier–Stokes Equation”, Journal of Functional Analysis, 149:1 (1997), 160  crossref
    23. A. V. Fursikov, O. Yu. Imanuvilov, “Local Exact Boundary Controllability of the Boussinesq Equation”, SIAM J Control Optim, 36:2 (1998), 391  crossref  mathscinet  zmath  isi  elib
    24. A. V. Fursikov, M. D. Gunzburger, L. S. Hou, “Boundary Value Problems and Optimal Boundary Control for the Navier–Stokes System: the Two-Dimensional Case”, SIAM J Control Optim, 36:3 (1998), 852  crossref  mathscinet  zmath  isi  elib
    25. L. S. Hou, S. S. Ravindran, “A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier–Stokes Equations”, SIAM J Control Optim, 36:5 (1998), 1795  crossref  mathscinet  zmath  isi  elib
    26. M. D. GUNZBURGER, L. S. HOU, S. MANSERVISIM, Y. Yan, “Computations of Optimal Controls for Incompressible Flows”, International Journal of Computational Fluid Dynamics, 11:1-2 (1998), 181  crossref  elib
    27. Fattorini H., Sritharan S., “Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion”, Appl. Math. Optim., 38:2 (1998), 159–192  crossref  mathscinet  zmath  isi
    28. Alekseev G., “Stationary Problems of Boundary Control for the Heat-Convection Equations”, Dokl. Akad. Nauk, 362:2 (1998), 174–177  mathnet  mathscinet  zmath  isi
    29. Barbu V., “Optimal Control of Navier–Stokes Equations with Periodic Inputs”, Nonlinear Anal.-Theory Methods Appl., 31:1-2 (1998), 15–31  crossref  mathscinet  zmath  isi
    30. M. D. Gunzburger, S. Manservisi, “The Velocity Tracking Problem for Navier–Stokes Flows with Bounded Distributed Controls”, SIAM J Control Optim, 37:6 (1999), 1913  crossref  mathscinet  zmath  isi
    31. A. A. Illarionov, “Asymptotics of solutions to the optimal control problem for time-independent Navier–Stokes equations”, Comput. Math. Math. Phys., 40:7 (2000), 1017–1026  mathnet  mathscinet  zmath  elib
    32. M.D. Gunzburger, S. Manservisi, “Analysis and approximation for linear feedback control for tracking the velocity in Navier–Stokes flows”, Computer Methods in Applied Mechanics and Engineering, 189:3 (2000), 803  crossref  elib
    33. Gunzburger M., Manservisi S., “The Velocity Tracking Problem for Navier–Stokes Flows with Boundary Control”, SIAM J. Control Optim., 39:2 (2000), 594–634  crossref  mathscinet  zmath  isi
    34. Gunzburger M., Manservisi S., “Analysis and Approximation of the Velocity Tracking Problem for Navier–Stokes Flows with Distributed Control”, SIAM J. Numer. Anal., 37:5 (2000), 1481–1512  crossref  mathscinet  zmath  isi
    35. Sritharan S., “Deterministic and Stochastic Control of Navier–Stokes Equation with Linear, Monotone, and Hyperviscosities”, Appl. Math. Optim., 41:2 (2000), 255–308  crossref  mathscinet  zmath  isi
    36. Gunzburger M., “Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows”, Flow Turbul. Combust., 65:3-4 (2000), 249–272  crossref  mathscinet  zmath  isi
    37. A. A. Illarionov, “Optimal Boundary Control of Steady-State Flow of a Viscous Inhomogeneous Incompressible Fluid”, Math. Notes, 69:5 (2001), 614–624  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    38. A. A. Illarionov, “Asymptotics of solutions to the optimal control problem for time-indepentent Navier–Stokes equations”, Comput. Math. Math. Phys., 41:7 (2001), 993–1004  mathnet  mathscinet  zmath
    39. Liu W., Yan N., “A Posteriori Error Estimates for Control Problems Governed by Stokes Equations”, SIAM J. Numer. Anal., 40:5 (2002), 1850–1869  crossref  mathscinet  zmath  isi
    40. Wang G., “Optimal Controls of 3-Dimensional Navier–Stokes Equations with State Constraints”, SIAM J. Control Optim., 41:2 (2002), 583–606  crossref  mathscinet  zmath  isi
    41. Wang G., “Pontryagin's Maximum Principle for Optimal Control of the Stationary Navier–Stokes Equations”, Nonlinear Anal.-Theory Methods Appl., 52:8 (2003), 1853–1866  crossref  mathscinet  zmath  isi
    42. Wang G., Wang L., “Maximum Principle of State-Constrained Optimal Control Governed by Fluid Dynamic Systems”, Nonlinear Anal.-Theory Methods Appl., 52:8 (2003), 1911–1931  crossref  mathscinet  zmath  isi
    43. A. V. Fursikov, M. D. Gunzburger, L. S. Hou, “Optimal Boundary Control for the Evolutionary Navier–Stokes System: The Three-Dimensional Case”, SIAM J Control Optim, 43:6 (2005), 2191  crossref  mathscinet  zmath  isi  elib
    44. Gejadze I.Yu., Copeland G.J.M., Navon I.M., “Open Boundary Control Problem for Navier–Stokes Equations Including a Free Surface: Data Assimilation”, Comput. Math. Appl., 52:8-9 (2006), 1269–1288  crossref  mathscinet  zmath  isi  elib
    45. Liu H., Yan N., “Global Superconvergence for Optimal Control Problems Governed by Stokes Equations”, Int. J. Numer. Anal. Model., 3:3 (2006), 283–302  mathscinet  zmath  isi
    46. A. Yu. Chebotarev, “Optimal control in nonstationary problems of magnetohydrodynamics”, J. Appl. Industr. Math., 3:2 (2009), 163–172  mathnet  crossref  mathscinet  elib
    47. Liu H., Yan N., “Recovery Type Superconvergence and a Posteriori Error Estimates for Control Problems Governed by Stokes Equations”, J. Comput. Appl. Math., 209:2 (2007), 187–207  crossref  mathscinet  zmath  adsnasa  isi
    48. T. Tachim Medjo, R. Temam, M. Ziane, “Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects”, Appl Mech Rev, 61:1 (2008), 010802  crossref  zmath  isi
    49. J. Appl. Industr. Math., 4:1 (2010), 6–18  mathnet  crossref  mathscinet
    50. A. Yu. Chebotarev, A. A. Illarionov, E. V. Amosova, “Variatsionnye neravenstva, kraevye zadachi i optimalnoe upravlenie dlya sistemy Nave–Stoksa”, Dalnevost. matem. zhurn., 8:1 (2008), 121–140  mathnet  elib
    51. Chebotarev A.Yu., “Optimal Control of Mhd-Flow Deceleration”, J. Appl. Mech. Tech. Phys., 49:5 (2008), 705–711  crossref  mathscinet  adsnasa  isi  elib
    52. V. G. Zvyagin, A. V. Kuznetsov, “Optimal control in a model of the motion of a viscoelastic medium with objective derivative”, Russian Math. (Iz. VUZ), 53:5 (2009), 48–53  mathnet  crossref  zmath  elib
    53. A. Yu. Chebotarev, “Impulse control of temperature in a free convection model”, Comput. Math. Math. Phys., 49:4 (2009), 594–601  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    54. A. V. Fursikov, “Flow of a viscous incompressible fluid around a body: boundary-value problems and minimization of the work of a fluid”, Journal of Mathematical Sciences, 180:6 (2012), 763–816  mathnet  crossref  mathscinet
    55. D. K. Potapov, “Upravlenie spektralnymi zadachami dlya uravnenii s razryvnymi operatorami”, Tr. IMM UrO RAN, 17, no. 1, 2011, 190–200  mathnet  elib
    56. V. G. Zvyagin, “Topological approximation approach to study of mathematical problems of hydrodynamics”, Journal of Mathematical Sciences, 201:6 (2014), 830–858  mathnet  crossref
    57. Deng Y., Liu Zh., Liu Y., Wu Y., “Combination of Topology Optimization and Optimal Control Method”, J. Comput. Phys., 257:A (2014), 374–399  crossref  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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