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March  2019, 24(3): 1115-1142. doi: 10.3934/dcdsb.2019009

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

Vladimir V. Chepyzhov 1,2,, , Anna Kostianko 3, and  Sergey Zelik 3,

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127051, Russian Federation
2. 

Voronezh State University, Universitetskaya sq. 1, Voronezh 394018, Russian Federation
3. 

University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom

* Corresponding author: V. V. Chepyzhov

To the blessed memory of Professor V. S. Melnik

Received  September 2017 Revised  March 2018 Published  March 2019 Early access  January 2019

Fund Project: The research of VVC was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037). The work of AK and SZ was partially supported by the EPSRC grant EP/P024920/1 and the work of SZ was partially supported by the Russian Foundation for Basic Research (projects 17-01-00515 and 18-01-00524).

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.

Bibliography: 38 titles.

Keywords: Hyperbolic relaxation, gap property, inertial manifold.
Mathematics Subject Classification: Primary: 35B40, 35B45.
Citation: Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

N. A. Chalkina, Sufficient condition for the existence of an inertial manifold for a hyperbolic equation with weak and strong dissipation, Russ. J. Math. Phys., 19 (2012), 11-20.  doi: 10.1134/S1061920812010025.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[4]

V. V. Chepyzhov, A. Yu. Goritsky and M. I. Vishik, Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation, Russ. J. Math. Phys., 12 (2005), 17-39.   Google Scholar

[5]

V. V. Chepyzhov and A. Yu. Goritsky, Global integral manifolds with exponential tracking for nonautonomous equations, Russ. J. Math. Phys., 5 (1997), 9-28.   Google Scholar

[6]

V. V. Chepyzhov and A. Yu. Goritsky, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 23-50.  doi: 10.1070/SM2005v196n04ABEH000889.  Google Scholar

[7]

A. Eden, S. V. Zelik and V. K. Kalantarov, Counterexamples to the regularity of Mané projections in the attractors theory, Russian Math. Surveys, 68 (2013), 199-226.  doi: 10.1070/rm2013v068n02abeh004828.  Google Scholar

[8]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[9]

P. Fabrie, C. Galushinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[10]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar

[11]

C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353.  doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin, New York, 1981.  Google Scholar

[13]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin, New York, 1977.  Google Scholar

[14]

V. Kalantarov, A. Savstianov and S. Zelik, Attractors for Damped Quintic Wave Equations in Bounded Domains, Annales Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y.  Google Scholar

[15]

N. Koksch, Almost sharp conditions for the existence of smooth inertial manifolds, in: Equadiff 9: Conference on Differential Equations and their Applications : Proceedings, edited by Z. Dosla, J. Kuben, J. Vosmansky, Masaryk University, Brno, 1998,139–166. Google Scholar

[16]

A. Kostianko and S. Zelik, Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions, Commun. Pure Appl. Anal., 16 (2017), 2357-2376.  doi: 10.3934/cpaa.2017116.  Google Scholar

[17]

A. Kostianko and S. Zelik, Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: Periodic bundary conditins, Commun. Pure Appl. Anal., 17 (2018), 285-317.  doi: 10.3934/cpaa.2018017.  Google Scholar

[18]

A. Kostianko and S. Zelik, Spatio-Temporal Averaging and Inertial Manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions, in preparation. Google Scholar

[19]

A. Kostianko and S. Zelik, Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions, Comm. Pure Appl. Anal., 14 (2015), 2069-2094.  doi: 10.3934/cpaa.2015.14.2069.  Google Scholar

[20]

A. Kostianko and S. Zelik, The Kwak transform and Inertial Manifolds Revisited, in preparation. Google Scholar

[21]

M. Kwak, Finite-dimensional inertial forms for 2D Navier-Stokes equations, Indiana Univ. Math. J., 41 (1992), 927-981.  doi: 10.1512/iumj.1992.41.41051.  Google Scholar

[22]

J. Mallet-Paret and G. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1 (1988), 805-866.  doi: 10.1090/S0894-0347-1988-0943276-7.  Google Scholar

[23]

J. Mallet-Paret, G. Sell and Z. Shao, Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055.  doi: 10.1512/iumj.1993.42.42048.  Google Scholar

[24]

M. Miklavcic, A sharp condition for existence of an inertial manifold, J. Dynam.Differential Equations, 3 (1991), 437-456.  doi: 10.1007/BF01049741.  Google Scholar

[25]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations. Vol. Ⅳ, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[26]

X. Mora and J. Solá-Morales, Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations, in: Dynamics of Infinite-Dimensional Systems (Lisbon, 1986), 187–210, NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 37, Springer, Berlin, 1987.  Google Scholar

[27]

X. Mora and J. Solá-Morales, Inertial manifolds of damped semilinear wave equations. Attractors, inertial manifolds and their approximation, in: (Marseille-Luminy, 1987). RAIRO Model. Math. Anal. Numer., 23 (1989), 489–505. doi: 10.1051/m2an/1989230304891.  Google Scholar

[28]

J. Robinson, Infinite-dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[29]

J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.  Google Scholar

[30]

A. Romanov, Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations, Russian Acad. Sci. Izv. Math., 43 (1994), 31-47.  doi: 10.1070/IM1994v043n01ABEH001557.  Google Scholar

[31]

A. Romanov, Three counterexamples in the theory of inertial manifolds, Math. Notes, 68 (2000), 378-385.  doi: 10.1007/BF02674562.  Google Scholar

[32]

R. Rosa and R. Temam, Inertial manifolds and normal hyperbolicity, Acta Appl. Math., 45 (1996), 1-50.  doi: 10.1007/BF00047882.  Google Scholar

[33]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[34]

R. Temam and S. Wang, Inertial forms of Navier-Stokes equations on the sphere, J. Funct. Anal., 117 (1993), 215-242.  doi: 10.1006/jfan.1993.1126.  Google Scholar

[35]

A. Yu. Goritskii and N. A. Chalkina, Inertial manifolds for weakly and strongly dissipative hyperbolic equations, J. Math. Sci., 197 (2014), 291-302.  doi: 10.1007/s10958-014-1715-4.  Google Scholar

[36]

S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proc. Royal Soc. Edinburgh, 144A (2014), 1245-1327.  doi: 10.1017/S0308210513000073.  Google Scholar

[37]

S. Zelik, Asymptotic Regularity of Solutions of Singularly Perturbed Damped Wave Equations with Supercritical Nonlinearities, Discrete Contin. Dyn. Syst., 11 (2004), 351-392.  doi: 10.3934/dcds.2004.11.351.  Google Scholar

[38]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and J. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing Ⅲ. Long-Time Behavior of Evolution Inclusions Solutions in Earth Data Analysis, Series: Advances in Mechanics and Mathematics, 27, Springer, Berlin, 2012. doi: 10.1007/978-3-642-28512-7.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

N. A. Chalkina, Sufficient condition for the existence of an inertial manifold for a hyperbolic equation with weak and strong dissipation, Russ. J. Math. Phys., 19 (2012), 11-20.  doi: 10.1134/S1061920812010025.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[4]

V. V. Chepyzhov, A. Yu. Goritsky and M. I. Vishik, Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation, Russ. J. Math. Phys., 12 (2005), 17-39.   Google Scholar

[5]

V. V. Chepyzhov and A. Yu. Goritsky, Global integral manifolds with exponential tracking for nonautonomous equations, Russ. J. Math. Phys., 5 (1997), 9-28.   Google Scholar

[6]

V. V. Chepyzhov and A. Yu. Goritsky, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 23-50.  doi: 10.1070/SM2005v196n04ABEH000889.  Google Scholar

[7]

A. Eden, S. V. Zelik and V. K. Kalantarov, Counterexamples to the regularity of Mané projections in the attractors theory, Russian Math. Surveys, 68 (2013), 199-226.  doi: 10.1070/rm2013v068n02abeh004828.  Google Scholar

[8]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[9]

P. Fabrie, C. Galushinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[10]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar

[11]

C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353.  doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin, New York, 1981.  Google Scholar

[13]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin, New York, 1977.  Google Scholar

[14]

V. Kalantarov, A. Savstianov and S. Zelik, Attractors for Damped Quintic Wave Equations in Bounded Domains, Annales Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y.  Google Scholar

[15]

N. Koksch, Almost sharp conditions for the existence of smooth inertial manifolds, in: Equadiff 9: Conference on Differential Equations and their Applications : Proceedings, edited by Z. Dosla, J. Kuben, J. Vosmansky, Masaryk University, Brno, 1998,139–166. Google Scholar

[16]

A. Kostianko and S. Zelik, Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions, Commun. Pure Appl. Anal., 16 (2017), 2357-2376.  doi: 10.3934/cpaa.2017116.  Google Scholar

[17]

A. Kostianko and S. Zelik, Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: Periodic bundary conditins, Commun. Pure Appl. Anal., 17 (2018), 285-317.  doi: 10.3934/cpaa.2018017.  Google Scholar

[18]

A. Kostianko and S. Zelik, Spatio-Temporal Averaging and Inertial Manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions, in preparation. Google Scholar

[19]

A. Kostianko and S. Zelik, Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions, Comm. Pure Appl. Anal., 14 (2015), 2069-2094.  doi: 10.3934/cpaa.2015.14.2069.  Google Scholar

[20]

A. Kostianko and S. Zelik, The Kwak transform and Inertial Manifolds Revisited, in preparation. Google Scholar

[21]

M. Kwak, Finite-dimensional inertial forms for 2D Navier-Stokes equations, Indiana Univ. Math. J., 41 (1992), 927-981.  doi: 10.1512/iumj.1992.41.41051.  Google Scholar

[22]

J. Mallet-Paret and G. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1 (1988), 805-866.  doi: 10.1090/S0894-0347-1988-0943276-7.  Google Scholar

[23]

J. Mallet-Paret, G. Sell and Z. Shao, Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055.  doi: 10.1512/iumj.1993.42.42048.  Google Scholar

[24]

M. Miklavcic, A sharp condition for existence of an inertial manifold, J. Dynam.Differential Equations, 3 (1991), 437-456.  doi: 10.1007/BF01049741.  Google Scholar

[25]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations. Vol. Ⅳ, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[26]

X. Mora and J. Solá-Morales, Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations, in: Dynamics of Infinite-Dimensional Systems (Lisbon, 1986), 187–210, NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 37, Springer, Berlin, 1987.  Google Scholar

[27]

X. Mora and J. Solá-Morales, Inertial manifolds of damped semilinear wave equations. Attractors, inertial manifolds and their approximation, in: (Marseille-Luminy, 1987). RAIRO Model. Math. Anal. Numer., 23 (1989), 489–505. doi: 10.1051/m2an/1989230304891.  Google Scholar

[28]

J. Robinson, Infinite-dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[29]

J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.  Google Scholar

[30]

A. Romanov, Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations, Russian Acad. Sci. Izv. Math., 43 (1994), 31-47.  doi: 10.1070/IM1994v043n01ABEH001557.  Google Scholar

[31]

A. Romanov, Three counterexamples in the theory of inertial manifolds, Math. Notes, 68 (2000), 378-385.  doi: 10.1007/BF02674562.  Google Scholar

[32]

R. Rosa and R. Temam, Inertial manifolds and normal hyperbolicity, Acta Appl. Math., 45 (1996), 1-50.  doi: 10.1007/BF00047882.  Google Scholar

[33]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[34]

R. Temam and S. Wang, Inertial forms of Navier-Stokes equations on the sphere, J. Funct. Anal., 117 (1993), 215-242.  doi: 10.1006/jfan.1993.1126.  Google Scholar

[35]

A. Yu. Goritskii and N. A. Chalkina, Inertial manifolds for weakly and strongly dissipative hyperbolic equations, J. Math. Sci., 197 (2014), 291-302.  doi: 10.1007/s10958-014-1715-4.  Google Scholar

[36]

S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proc. Royal Soc. Edinburgh, 144A (2014), 1245-1327.  doi: 10.1017/S0308210513000073.  Google Scholar

[37]

S. Zelik, Asymptotic Regularity of Solutions of Singularly Perturbed Damped Wave Equations with Supercritical Nonlinearities, Discrete Contin. Dyn. Syst., 11 (2004), 351-392.  doi: 10.3934/dcds.2004.11.351.  Google Scholar

[38]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and J. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing Ⅲ. Long-Time Behavior of Evolution Inclusions Solutions in Earth Data Analysis, Series: Advances in Mechanics and Mathematics, 27, Springer, Berlin, 2012. doi: 10.1007/978-3-642-28512-7.  Google Scholar

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