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doi: 10.3934/eect.2021049
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Sufficient conditions for the continuity of inertial manifolds for singularly perturbed problems

Department of Mathematics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia

*Tel.:+966 138607263; fax:+966 138602340

Received  August 2021 Revised  June 2021 Early access September 2021

We consider a nonlinear evolution equation in the form
$ {{\rm{U}}_t} + {{\rm{A}}_\varepsilon }{\rm{U}} + {{\rm{N}}_\varepsilon }{{\rm{G}}_\varepsilon }({\rm{U}}) = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{E}}_\varepsilon }} \right)$
together with its singular limit problem as
$ \varepsilon\to 0 $
$ \begin{align*} U_t+{\rm A} U+ {\rm N}{\rm G}(U) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{E}} }} \right)\end{align*} $
where
$ \varepsilon\in (0,1] $
(possibly
$ \varepsilon = 0 $
),
$ {\rm A}_\varepsilon $
and
$ {\rm A} $
are linear closed (possibly) unbounded operators,
$ {\rm N}_\varepsilon $
and
$ {\rm N} $
are linear (possibly) unbounded operators,
$ {\rm G}_\varepsilon $
and
$ {\rm G} $
are nonlinear functions. We give sufficient conditions on
$ {\rm A}_\varepsilon, $
$ {\rm N}_\varepsilon $
and
$ {\rm G}_\varepsilon $
(and also
$ {\rm A} $
,
$ {\rm N} $
and
$ {\rm G} $
) that guarantee not only the existence of an inertial manifold of dimension independent of
$ \varepsilon $
for (Eε) on a Banach space
$ {\mathcal H} $
, but also the Hölder continuity, lower and upper semicontinuity at
$ \varepsilon = 0 $
of the intersection of the inertial manifold with a bounded absorbing set. Applications to higher order viscous Cahn-Hilliard-Oono equations, the hyperbolic type equations and the phase-field systems, subject to either Neumann or Dirichlet boundary conditions (BC) (in which case
$ \Omega\subset{\mathbb R}^d $
is a bounded domain with smooth boundary) or periodic BC (in which case
$ \Omega = \Pi_{i = 1}^d(0,L_i), $
$ L_i>0 $
),
$ d = 1,2\; {\rm or} \;3$
, are considered. These three classes of dissipative equations read
$ \begin{align*} \phi_{t}+N(\varepsilon \phi_t+N^{\alpha+1} \phi +N\phi + g(\phi))+\sigma\phi = 0,\quad\alpha\in\mathbb N, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{P}}_\varepsilon }} \right)\end{align*} $
$ \begin{align*} \varepsilon \phi_{tt}+\phi_{t}+N^\alpha(N \phi + g(\phi))+ \sigma\phi = 0,\quad\alpha = 0, 1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{H}}_\varepsilon }} \right) \end{align*} $
and
$ \begin{align*} \left\{\begin{aligned} & \phi_{t}+N^\alpha (N \phi + g(\phi)-u)+\sigma\phi = 0,\\&\varepsilon u_t+\phi_t+N u = 0,\end{aligned}\right.\quad\alpha = 0, 1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right) \end{align*} $
respectively, where
$ \sigma\ge 0 $
and the Laplace operator is defined as
$ N = -\Delta:{\mathscr D}(N) = \{\psi\in H^2(\Omega),\,\psi\,{\rm subject \,\,to \,\,the\,\, BC}\}\to \dot L^2(\Omega)\,\,{\rm or}\,\,L^2(\Omega). $
We assume that, for a given real number
$ {\frak c}_1>0, $
there exists a positive integer
$ n = n({\frak c}_1) $
such that
$ \lambda_{n+1}-\lambda_n>{\frak c}_1 $
, where
$ \{\lambda_k\}_{k\in\mathbb N^*} $
are the eigenvalues of
$ N $
. There exists a real number
$ {\mathscr M}>0 $
such that the nonlinear function
$ g: V_j\to V_j $
satisfies the conditions
$ \|g(\psi)\|_j\le {\mathscr M} $
and
$ \|g(\psi)-g(\varphi)\|_{V_j}\le {\mathscr M}\|\psi-\varphi\|_{V_j} $
,
$ \forall\psi,\varphi\in V_j $
, where
$ V_j = {\mathscr D}(N^{j/2}) $
,
$ j = 1 $
for Problems (Pε) and (Sε) and
$ j = 0, 2\alpha $
for Problem (Hε). We further require
$ g\in{\mathcal C}^1( V_j, V_j) $
,
$ \|g'(\psi)\varphi\|_j\le {\mathscr M}\|\varphi\|_j $
for Problems (Hε) and (Sε).
Citation: Ahmed Bonfoh. Sufficient conditions for the continuity of inertial manifolds for singularly perturbed problems. Evolution Equations and Control Theory, doi: 10.3934/eect.2021049
References:
[1]

A. Bonfoh, Existence and continuity of inertial manifolds for the hyperbolic relaxation of the viscous Cahn-Hilliard equation, Appl. Math Optim., 84 (2021), 3339-3416.  doi: 10.1007/s00245-021-09749-9.

[2]

A. Bonfoh, Dynamics of a conserved phase-field system, Appl. Anal., 95 (2016), 44-62.  doi: 10.1080/00036811.2014.997225.

[3]

A. Bonfoh, The singular limit dynamics of the phase-field equations, Ann. Mat. Pura Appl., 190 (2011), 105-144.  doi: 10.1007/s10231-010-0141-6.

[4]

A. Bonfoh and C. D. Enyi, The Cahn-Hilliard equation as limit of a conserved phase-field system, Asymptot. Anal., 101 (2017), 97-148.  doi: 10.3233/ASY-161395.

[5]

A. BonfohM. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited, Nonlinear Differ. Equ. Appli., 17 (2010), 663-695.  doi: 10.1007/s00030-010-0075-0.

[6]

V. ChepyzhovA. Kostianko and S. Zelik, Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1115-1142.  doi: 10.3934/dcdsb.2019009.

[7]

L. CherfilsA. MiranvilleS. Peng and W. Zhang, Higher-order generalized Cahn-Hilliard equations, Electron. J. Qual. Theory Differ. Equ., 9 (2017), 1-22.  doi: 10.14232/ejqtde.2017.1.9.

[8]

L. CherfilsA. Miranville and S. Peng, Higher-order models in phase separation, J. Appl. Anal. Comput., 7 (2017), 39-56.  doi: 10.11948/2017003.

[9]

L. CherfilsA. Miranville and S. Peng, Higher-order anisotropic models in phase separation, Adv. Nonlinear Anal., 8 (2019), 278-302.  doi: 10.1515/anona-2016-0137.

[10] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278. Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.
[11]

S.-N. Chow and K. Lu, Invariant manifolds for flow in Banach spaces, J. Diff. Eqns, 74 (1988), 285-317.  doi: 10.1016/0022-0396(88)90007-1.

[12]

S.-N. ChowK. Lu and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312.  doi: 10.1016/0022-247X(92)90115-T.

[13]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; http://www.emis.de/monographs/Chueshov/.

[14]

A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymptotic Anal., 4 (1991), 161-185.  doi: 10.3233/ASY-1991-4202.

[15]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.

[16]

C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differential Equations, 128 (1996), 387-414.  doi: 10.1006/jdeq.1996.0101.

[17]

C. FoiasG. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-355.  doi: 10.1016/0022-0396(88)90110-6.

[18]

S. GattiM. GrasselliA. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation, J. Math. Anal. Appl., 312 (2005), 230-247.  doi: 10.1016/j.jmaa.2005.03.029.

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.

[20]

J. K. Hale and G. Raugel, Upper-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.

[21]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, Heidelberg, New York, 1981.

[22]

H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346.  doi: 10.2140/pjm.1966.19.285.

[23]

J. Mallet-Paret and G. R. 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.

[24]

A. J. Milani and N. J. Koksch, An Introduction to Semiflows, , Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 134. Chapman & Hall/CRC, Boca Raton, FL, 2005.

[25]

X. Mora and J. Solà-Morales, The singular limit dynamics of semilinear damped wave equations, J. Differential Equations, 78 (1989), 262-307.  doi: 10.1016/0022-0396(89)90065-X.

[26]

B. NicolaenkoB. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297.  doi: 10.1080/03605308908820597.

[27]

L. E. PayneG. Polya and H.F. Weinberger, On the ratio of consecutive eigenvalue, J. Math. and Phys., 35 (1956), 289-298.  doi: 10.1002/sapm1956351289.

[28]

M. Prizzi and K. P. Rybakowski, On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains, Studia Math., 154 (2003), 253-275.  doi: 10.4064/sm154-3-6.

[29]

M. PrizziM. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Studia Math., 151 (2002), 109-140.  doi: 10.4064/sm151-2-2.

[30]

I. Richards, On the gaps between numbers which are sums of two squares, Adv. in Math., 46 (1982), 1-2.  doi: 10.1016/0001-8708(82)90051-2.

[31]

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

[32] J. C. 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. 
[33]

A. Savostianov and S. Zelik, Global well-posedness and attractors for the hyperbolic Cahn-Hilliard-Oono equation in the whole space, Math. Models Methods Appl. Sci., 26 (2016), 1357-1384.  doi: 10.1142/S0218202516500329.

[34]

A. Savostianov and S. Zelik, Finite dimensionality of the attractor for the hyperbolic Cahn-Hilliard-Oono equation in $\mathbb R^3$, Math. Methods Appl. Sci., 39 (2016), 1254-1267.  doi: 10.1002/mma.3569.

[35]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, Berlin, Heidelberg, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[36]

D. Selcovic, Smoothness of the singular limit of inertial manifolds of singularly perturbed evolution equations, Nonlinear Anal., 28 (1997), 199-215.  doi: 10.1016/0362-546X(95)00139-M.

[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on the unit sphere, Comm. Math. Phys., 285 (2009), 975-990.  doi: 10.1007/s00220-008-0460-2.

[39]

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

[40]

S. Zheng and A. Milani, Global attractors for singular perturbations of the Cahn-Hilliard equations, J. Differential Equations, 209 (2005), 101-139.  doi: 10.1016/j.jde.2004.08.026.

[41]

S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations, Nonlinear Anal., 57 (2004), 843-877.  doi: 10.1016/j.na.2004.03.023.

show all references

References:
[1]

A. Bonfoh, Existence and continuity of inertial manifolds for the hyperbolic relaxation of the viscous Cahn-Hilliard equation, Appl. Math Optim., 84 (2021), 3339-3416.  doi: 10.1007/s00245-021-09749-9.

[2]

A. Bonfoh, Dynamics of a conserved phase-field system, Appl. Anal., 95 (2016), 44-62.  doi: 10.1080/00036811.2014.997225.

[3]

A. Bonfoh, The singular limit dynamics of the phase-field equations, Ann. Mat. Pura Appl., 190 (2011), 105-144.  doi: 10.1007/s10231-010-0141-6.

[4]

A. Bonfoh and C. D. Enyi, The Cahn-Hilliard equation as limit of a conserved phase-field system, Asymptot. Anal., 101 (2017), 97-148.  doi: 10.3233/ASY-161395.

[5]

A. BonfohM. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited, Nonlinear Differ. Equ. Appli., 17 (2010), 663-695.  doi: 10.1007/s00030-010-0075-0.

[6]

V. ChepyzhovA. Kostianko and S. Zelik, Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1115-1142.  doi: 10.3934/dcdsb.2019009.

[7]

L. CherfilsA. MiranvilleS. Peng and W. Zhang, Higher-order generalized Cahn-Hilliard equations, Electron. J. Qual. Theory Differ. Equ., 9 (2017), 1-22.  doi: 10.14232/ejqtde.2017.1.9.

[8]

L. CherfilsA. Miranville and S. Peng, Higher-order models in phase separation, J. Appl. Anal. Comput., 7 (2017), 39-56.  doi: 10.11948/2017003.

[9]

L. CherfilsA. Miranville and S. Peng, Higher-order anisotropic models in phase separation, Adv. Nonlinear Anal., 8 (2019), 278-302.  doi: 10.1515/anona-2016-0137.

[10] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278. Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.
[11]

S.-N. Chow and K. Lu, Invariant manifolds for flow in Banach spaces, J. Diff. Eqns, 74 (1988), 285-317.  doi: 10.1016/0022-0396(88)90007-1.

[12]

S.-N. ChowK. Lu and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312.  doi: 10.1016/0022-247X(92)90115-T.

[13]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; http://www.emis.de/monographs/Chueshov/.

[14]

A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymptotic Anal., 4 (1991), 161-185.  doi: 10.3233/ASY-1991-4202.

[15]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.

[16]

C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differential Equations, 128 (1996), 387-414.  doi: 10.1006/jdeq.1996.0101.

[17]

C. FoiasG. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-355.  doi: 10.1016/0022-0396(88)90110-6.

[18]

S. GattiM. GrasselliA. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation, J. Math. Anal. Appl., 312 (2005), 230-247.  doi: 10.1016/j.jmaa.2005.03.029.

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.

[20]

J. K. Hale and G. Raugel, Upper-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.

[21]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, Heidelberg, New York, 1981.

[22]

H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346.  doi: 10.2140/pjm.1966.19.285.

[23]

J. Mallet-Paret and G. R. 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.

[24]

A. J. Milani and N. J. Koksch, An Introduction to Semiflows, , Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 134. Chapman & Hall/CRC, Boca Raton, FL, 2005.

[25]

X. Mora and J. Solà-Morales, The singular limit dynamics of semilinear damped wave equations, J. Differential Equations, 78 (1989), 262-307.  doi: 10.1016/0022-0396(89)90065-X.

[26]

B. NicolaenkoB. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297.  doi: 10.1080/03605308908820597.

[27]

L. E. PayneG. Polya and H.F. Weinberger, On the ratio of consecutive eigenvalue, J. Math. and Phys., 35 (1956), 289-298.  doi: 10.1002/sapm1956351289.

[28]

M. Prizzi and K. P. Rybakowski, On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains, Studia Math., 154 (2003), 253-275.  doi: 10.4064/sm154-3-6.

[29]

M. PrizziM. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Studia Math., 151 (2002), 109-140.  doi: 10.4064/sm151-2-2.

[30]

I. Richards, On the gaps between numbers which are sums of two squares, Adv. in Math., 46 (1982), 1-2.  doi: 10.1016/0001-8708(82)90051-2.

[31]

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

[32] J. C. 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. 
[33]

A. Savostianov and S. Zelik, Global well-posedness and attractors for the hyperbolic Cahn-Hilliard-Oono equation in the whole space, Math. Models Methods Appl. Sci., 26 (2016), 1357-1384.  doi: 10.1142/S0218202516500329.

[34]

A. Savostianov and S. Zelik, Finite dimensionality of the attractor for the hyperbolic Cahn-Hilliard-Oono equation in $\mathbb R^3$, Math. Methods Appl. Sci., 39 (2016), 1254-1267.  doi: 10.1002/mma.3569.

[35]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, Berlin, Heidelberg, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[36]

D. Selcovic, Smoothness of the singular limit of inertial manifolds of singularly perturbed evolution equations, Nonlinear Anal., 28 (1997), 199-215.  doi: 10.1016/0362-546X(95)00139-M.

[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on the unit sphere, Comm. Math. Phys., 285 (2009), 975-990.  doi: 10.1007/s00220-008-0460-2.

[39]

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

[40]

S. Zheng and A. Milani, Global attractors for singular perturbations of the Cahn-Hilliard equations, J. Differential Equations, 209 (2005), 101-139.  doi: 10.1016/j.jde.2004.08.026.

[41]

S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations, Nonlinear Anal., 57 (2004), 843-877.  doi: 10.1016/j.na.2004.03.023.

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