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2016 | OriginalPaper | Buchkapitel

15. Evolutionary Systems and the Navier–Stokes Equations

verfasst von : Grzegorz Łukaszewicz, Piotr Kalita

Erschienen in: Navier–Stokes Equations

Verlag: Springer International Publishing

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Abstract

This chapter is devoted to the study of three-dimensional nonstationary Navier–Stokes equations with the multivalued frictional boundary condition. We use the formalism of evolutionary systems to prove the existence of weak global attractor for the studied problem.

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Literatur
9.
Zurück zum Zitat A.V. Babin, M.I. Vishik, Maximal attractors of semigroups corresponding to evolution differential equations. Math. USSR Sb. 54, 387–408 (1986)CrossRefMATH A.V. Babin, M.I. Vishik, Maximal attractors of semigroups corresponding to evolution differential equations. Math. USSR Sb. 54, 387–408 (1986)CrossRefMATH
11.
Zurück zum Zitat J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. Nonlinear Sci. 7, 475–502 (1997). Erratum, ibid 8 (1998) 233. Corrected version appears in “Mechanics: from Theory to Computation” (Springer, Berlin, 2000), pp. 447–474 J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. Nonlinear Sci. 7, 475–502 (1997). Erratum, ibid 8 (1998) 233. Corrected version appears in “Mechanics: from Theory to Computation” (Springer, Berlin, 2000), pp. 447–474
32.
Zurück zum Zitat L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)MathSciNetCrossRefMATH L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)MathSciNetCrossRefMATH
34.
Zurück zum Zitat C.S. Cao, E.S. Titi, Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57, 2643–2661 (2008)MathSciNetCrossRefMATH C.S. Cao, E.S. Titi, Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57, 2643–2661 (2008)MathSciNetCrossRefMATH
40.
Zurück zum Zitat T. Caraballo, P. Marín-Rubio, J.C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour. Set-Valued Anal. 11, 297–322 (2003)MathSciNetCrossRefMATH T. Caraballo, P. Marín-Rubio, J.C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour. Set-Valued Anal. 11, 297–322 (2003)MathSciNetCrossRefMATH
45.
Zurück zum Zitat T. Caraballo, G. Łukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal. Theory 64, 484–498 (2006)MathSciNetCrossRefMATH T. Caraballo, G. Łukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal. Theory 64, 484–498 (2006)MathSciNetCrossRefMATH
47.
Zurück zum Zitat T. Caraballo, P.E. Kloeden, J. Real, Invariant measures and statistical solutions of the globally modified Navier–Stokes equations. Discrete Cont. Dyn. B 10, 761–781 (2008)MathSciNetCrossRefMATH T. Caraballo, P.E. Kloeden, J. Real, Invariant measures and statistical solutions of the globally modified Navier–Stokes equations. Discrete Cont. Dyn. B 10, 761–781 (2008)MathSciNetCrossRefMATH
48.
Zurück zum Zitat T. Caraballo, J. Real, A.M. Márquez, Three-dimensional system of globally modified Navier–Stokes equations with delay. Int. J. Bifurcation Chaos 20, 2869–2883 (2010)MathSciNetCrossRefMATH T. Caraballo, J. Real, A.M. Márquez, Three-dimensional system of globally modified Navier–Stokes equations with delay. Int. J. Bifurcation Chaos 20, 2869–2883 (2010)MathSciNetCrossRefMATH
59.
Zurück zum Zitat V.V. Chepyzhov, M.I. Vishik, Trajectory attractors for evolution equations. C. R. Acad. Sci. I Math. 321, 1309–1314 (1995)MathSciNetMATH V.V. Chepyzhov, M.I. Vishik, Trajectory attractors for evolution equations. C. R. Acad. Sci. I Math. 321, 1309–1314 (1995)MathSciNetMATH
63.
Zurück zum Zitat V.V. Chepyzhov, E.S. Titi, M.I. Vishik, On the convergence of solutions of the Leray-alpha model to the trajectory attractor of the 3D Navier-Stokes system. Discrete Cont. Dyn. Syst. 17, 481–500 (2007)MathSciNetMATH V.V. Chepyzhov, E.S. Titi, M.I. Vishik, On the convergence of solutions of the Leray-alpha model to the trajectory attractor of the 3D Navier-Stokes system. Discrete Cont. Dyn. Syst. 17, 481–500 (2007)MathSciNetMATH
64.
Zurück zum Zitat S.I. Chernyshenko, P. Constantin, J.C. Robinson, E.S. Titi, A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations. J. Math. Phys. 48 (2007). Article ID: 065204 S.I. Chernyshenko, P. Constantin, J.C. Robinson, E.S. Titi, A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations. J. Math. Phys. 48 (2007). Article ID: 065204
66.
68.
Zurück zum Zitat A. Cheskidov, S. Friedlander, R. Shvydkoy, On the energy equality for weak solutions of the 3D Navier–Stokes equations, in Advances in Mathematical Fluid Mechanics, ed. by R. Rannacher, A. Sequeira, Springer Berlin, Heidelberg (2010), pp. 171–175 A. Cheskidov, S. Friedlander, R. Shvydkoy, On the energy equality for weak solutions of the 3D Navier–Stokes equations, in Advances in Mathematical Fluid Mechanics, ed. by R. Rannacher, A. Sequeira, Springer Berlin, Heidelberg (2010), pp. 171–175
73.
Zurück zum Zitat P. Constantin, C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier–Stokes equations. Indiana Univ. Math. J. 42, 775–789 (1993)MathSciNetCrossRefMATH P. Constantin, C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier–Stokes equations. Indiana Univ. Math. J. 42, 775–789 (1993)MathSciNetCrossRefMATH
82.
Zurück zum Zitat M. Dashti, J.C. Robinson, A simple proof of uniqueness of the particle trajectories for solutions of the Navier–Stokes equations. Nonlinearity 22, 735–746 (2009)MathSciNetCrossRefMATH M. Dashti, J.C. Robinson, A simple proof of uniqueness of the particle trajectories for solutions of the Navier–Stokes equations. Nonlinearity 22, 735–746 (2009)MathSciNetCrossRefMATH
86.
Zurück zum Zitat T. Dłotko, New look at the Navier–Stokes equation (2015). arXiv:1501.02085 T. Dłotko, New look at the Navier–Stokes equation (2015). arXiv:1501.02085
98.
Zurück zum Zitat C. Foiaş, R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87, 359–369 (1989)MathSciNetCrossRefMATH C. Foiaş, R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87, 359–369 (1989)MathSciNetCrossRefMATH
99.
Zurück zum Zitat C. Foiaş, O.P. Manley, R. Rosa, R. Temam, Navier-Stokes Equations and Turbulence (Cambridge University Press, Cambridge, 2001)MATH C. Foiaş, O.P. Manley, R. Rosa, R. Temam, Navier-Stokes Equations and Turbulence (Cambridge University Press, Cambridge, 2001)MATH
100.
Zurück zum Zitat C. Foiaş, D.D. Holm, E.S. Titi, The Navier–Stokes-alpha model of fluid turbulence. Phys. D 152, 505–519 (2001)MathSciNetMATH C. Foiaş, D.D. Holm, E.S. Titi, The Navier–Stokes-alpha model of fluid turbulence. Phys. D 152, 505–519 (2001)MathSciNetMATH
102.
Zurück zum Zitat C. Foiaş, R. Rosa, R. Temam, Topological properties of the weak global attractor of the three-dimensional Navier–Stokes equations. Discrete Cont. Dyn. Syst. 27, 1611–1631 (2010)MathSciNetCrossRefMATH C. Foiaş, R. Rosa, R. Temam, Topological properties of the weak global attractor of the three-dimensional Navier–Stokes equations. Discrete Cont. Dyn. Syst. 27, 1611–1631 (2010)MathSciNetCrossRefMATH
115.
128.
Zurück zum Zitat O.V. Kapustyan, J. Valero, Weak and strong attractors for the 3D Navier–Stokes system. J. Differ. Equ. 240, 249–278 (2007)MathSciNetCrossRefMATH O.V. Kapustyan, J. Valero, Weak and strong attractors for the 3D Navier–Stokes system. J. Differ. Equ. 240, 249–278 (2007)MathSciNetCrossRefMATH
129.
Zurück zum Zitat O.V. Kapustyan, J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions. Int. J. Bifurcation Chaos 20, 2723–2734 (2010)MathSciNetCrossRefMATH O.V. Kapustyan, J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions. Int. J. Bifurcation Chaos 20, 2723–2734 (2010)MathSciNetCrossRefMATH
138.
Zurück zum Zitat P.E. Kloeden, J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier–Stokes equations. Proc. R. Soc. Lond. Ser. A 463, 1491–1508 (2007)MathSciNetCrossRefMATH P.E. Kloeden, J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier–Stokes equations. Proc. R. Soc. Lond. Ser. A 463, 1491–1508 (2007)MathSciNetCrossRefMATH
139.
Zurück zum Zitat P.E. Kloeden, J.A. Langa, J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier–Stokes equations. Commun. Pure Appl. Anal. 6, 937–955 (2007)MathSciNetCrossRefMATH P.E. Kloeden, J.A. Langa, J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier–Stokes equations. Commun. Pure Appl. Anal. 6, 937–955 (2007)MathSciNetCrossRefMATH
141.
Zurück zum Zitat P.E. Kloeden, P. Marín-Rubio, J. Valero, The envelope attractor of non-strict multivalued dynamical systems with application to the 3D Navier–Stokes and reaction-diffusion equations. Set-Valued Var. Anal. 21, 517–540 (2013)MathSciNetCrossRefMATH P.E. Kloeden, P. Marín-Rubio, J. Valero, The envelope attractor of non-strict multivalued dynamical systems with application to the 3D Navier–Stokes and reaction-diffusion equations. Set-Valued Var. Anal. 21, 517–540 (2013)MathSciNetCrossRefMATH
143.
144.
Zurück zum Zitat I. Kukavica, M. Ziane, Navier–Stokes equations with regularity in one direction. J. Math. Phys. 48 (2007). Article ID: 065203 I. Kukavica, M. Ziane, Navier–Stokes equations with regularity in one direction. J. Math. Phys. 48 (2007). Article ID: 065203
154.
167.
Zurück zum Zitat J. Málek, J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids. J. Differ Equ. 127, 498–518 (1996)MathSciNetCrossRefMATH J. Málek, J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids. J. Differ Equ. 127, 498–518 (1996)MathSciNetCrossRefMATH
171.
Zurück zum Zitat V.S. Melnik, J. Valero, On attractors of multivalued semiflows and differential inclusions. Set-Valued Anal. 6, 83–111 (1998)MathSciNetCrossRef V.S. Melnik, J. Valero, On attractors of multivalued semiflows and differential inclusions. Set-Valued Anal. 6, 83–111 (1998)MathSciNetCrossRef
172.
Zurück zum Zitat V.S. Melnik, J. Valero, Addendum to “On Attractors of Multivalued Semiflows and Differential Inclusions” [Set-Valued Anal. 6, 83–111 (1998)]. Set-Valued Anal. 16, 507–509 (2008) V.S. Melnik, J. Valero, Addendum to “On Attractors of Multivalued Semiflows and Differential Inclusions” [Set-Valued Anal. 6, 83–111 (1998)]. Set-Valued Anal. 16, 507–509 (2008)
189.
Zurück zum Zitat V. Pata, A. Miranville, On the regularity of solutions to the Navier–Stokes equations. Commun. Pure Appl. Anal. 11, 747–761 (2012)MathSciNetCrossRefMATH V. Pata, A. Miranville, On the regularity of solutions to the Navier–Stokes equations. Commun. Pure Appl. Anal. 11, 747–761 (2012)MathSciNetCrossRefMATH
200.
Zurück zum Zitat J.C. Robinson, J.L. Rodrigo, W. Sadowski, Classical theory of the three-dimensional Navier-Stokes equations (2016, to appear) J.C. Robinson, J.L. Rodrigo, W. Sadowski, Classical theory of the three-dimensional Navier-Stokes equations (2016, to appear)
202.
Zurück zum Zitat R. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier–Stokes equations. J. Differ. Equ. 229, 257–269 (2006)MathSciNetCrossRefMATH R. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier–Stokes equations. J. Differ. Equ. 229, 257–269 (2006)MathSciNetCrossRefMATH
205.
Zurück zum Zitat V. Scheffer, Turbulence and Hausdorff dimension, in Turbulence and the Navier–Stokes Equations, ed. by R. Temam. Lecture Notes in Mathematics, vol. 565 (1976), Springer, Berlin, New York pp. 94–112 V. Scheffer, Turbulence and Hausdorff dimension, in Turbulence and the Navier–Stokes Equations, ed. by R. Temam. Lecture Notes in Mathematics, vol. 565 (1976), Springer, Berlin, New York pp. 94–112
210.
219.
Zurück zum Zitat R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, 3rd revised edn. (North-Holland, Amsterdam, New York, Oxford, 1984) R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, 3rd revised edn. (North-Holland, Amsterdam, New York, Oxford, 1984)
Metadaten
Titel
Evolutionary Systems and the Navier–Stokes Equations
verfasst von
Grzegorz Łukaszewicz
Piotr Kalita
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-27760-8_15

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