June  2012, 2(2): 141-170. doi: 10.3934/mcrf.2012.2.141

The simplest semilinear parabolic equation of normal type

1. 

Department of Mechanics & Mathematics, Moscow State University, Moscow 119991, Russian Federation

Received  June 2011 Revised  February 2012 Published  May 2012

The notion of semilinear parabolic equation of normal type is introduced. The structure of dynamical flow corresponding to equation of this type with periodic boundary condition is investigated. Stabilization of mentioned equation with arbitrary initial condition by start control supported in prescribed subset is constructed.
Citation: Andrei Fursikov. The simplest semilinear parabolic equation of normal type. Mathematical Control & Related Fields, 2012, 2 (2) : 141-170. doi: 10.3934/mcrf.2012.2.141
References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control," Translated from the Russian by V. M. Volosov, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987.  Google Scholar

[2]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, in "Séminaire sur les Équations aux Dérivées Partielles, 1993-1994," Exp. No. VIII, École Polytechnique, Palaiseau, (1994), 12 pp.  Google Scholar

[3]

J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Annals of Mathematics (2), 173 (2011), 983-1012. doi: 10.4007/annals.2011.173.2.9.  Google Scholar

[4]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[5]

A. V. Fursikov, Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations, Discrete and Continuous Dynamical System Ser. S, 3 (2010), 269-289.  Google Scholar

[6]

A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications," Translations of Mathematical Monographs, 187, Amer. Math. Society, Providence, Rhode Island, 2000.  Google Scholar

[7]

A. V. Fursikov, Stabilizability of a quasilinear parabolic equation by means of boundary feedback control, Sbornik: Mathematics, 192 (2001), 593-639. doi: 10.1070/SM2001v192n04ABEH000560.  Google Scholar

[8]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. of Math. Fluid Mech., 3 (2001), 259-301.  Google Scholar

[9]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314. doi: 10.3934/dcds.2004.10.289.  Google Scholar

[10]

A. V. Fursikov, Unique solvability "in large" of the three-dimensional Navier-Stokes system and moment equations for a dense set of data, in "Mathematical Problems of Statistical Hydrodynamics" (by M. I. Vishik and A. V. Fursikov), Appendix 1, Kluwer, Dordrecht, 1988. Google Scholar

[11]

M. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Mathematics, 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[12]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, revixed and enlarged, Translated from the Russian by Richard A. Silverman and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[13]

J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace , Acta Matematica, 63 (1933), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[14]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'Hydrodynamique, Journal de Mathématiques Pures et Appliquées, 12 (1933), 1-82. Google Scholar

[15]

F. Weissler, The Navier-Stokes initial value problem in $L^p$, Archiv for Rational Mechanics and Analysis, 74 (1980), 219-230. doi: 10.1007/BF00280539.  Google Scholar

show all references

References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control," Translated from the Russian by V. M. Volosov, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987.  Google Scholar

[2]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, in "Séminaire sur les Équations aux Dérivées Partielles, 1993-1994," Exp. No. VIII, École Polytechnique, Palaiseau, (1994), 12 pp.  Google Scholar

[3]

J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Annals of Mathematics (2), 173 (2011), 983-1012. doi: 10.4007/annals.2011.173.2.9.  Google Scholar

[4]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[5]

A. V. Fursikov, Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations, Discrete and Continuous Dynamical System Ser. S, 3 (2010), 269-289.  Google Scholar

[6]

A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications," Translations of Mathematical Monographs, 187, Amer. Math. Society, Providence, Rhode Island, 2000.  Google Scholar

[7]

A. V. Fursikov, Stabilizability of a quasilinear parabolic equation by means of boundary feedback control, Sbornik: Mathematics, 192 (2001), 593-639. doi: 10.1070/SM2001v192n04ABEH000560.  Google Scholar

[8]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. of Math. Fluid Mech., 3 (2001), 259-301.  Google Scholar

[9]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314. doi: 10.3934/dcds.2004.10.289.  Google Scholar

[10]

A. V. Fursikov, Unique solvability "in large" of the three-dimensional Navier-Stokes system and moment equations for a dense set of data, in "Mathematical Problems of Statistical Hydrodynamics" (by M. I. Vishik and A. V. Fursikov), Appendix 1, Kluwer, Dordrecht, 1988. Google Scholar

[11]

M. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Mathematics, 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[12]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, revixed and enlarged, Translated from the Russian by Richard A. Silverman and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[13]

J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace , Acta Matematica, 63 (1933), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[14]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'Hydrodynamique, Journal de Mathématiques Pures et Appliquées, 12 (1933), 1-82. Google Scholar

[15]

F. Weissler, The Navier-Stokes initial value problem in $L^p$, Archiv for Rational Mechanics and Analysis, 74 (1980), 219-230. doi: 10.1007/BF00280539.  Google Scholar

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