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November  2012, 11(6): 2393-2416. doi: 10.3934/cpaa.2012.11.2393

Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems

Maurizio Grasselli 1, and  Morgan Pierre 2,

1. 

Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano
2. 

Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil

Received  October 2010 Revised  December 2010 Published  April 2012

Following a result of Chill and Jendoubi in the continuous case, we study the asymptotic behavior of sequences $(U^n)_n$ in $R^d$ which satisfy the following backward Euler scheme:

$\varepsilon\frac{(U^{n+1}-2U^n+U^{n-1}}{\Delta t^2} +\frac{U^{n+1}-U^n}{\Delta t}+\nabla F(U^{n+1})=G^{n+1}, n\ge 0, $

where $\Delta t>0$ is the time step, $\varepsilon\ge 0$, $(G^{n+1})_n$ is a sequence in $ R^d$ which converges to $0$ in a suitable way, and $F\in C^{1,1}_{l o c}(R^d, R)$ is a function which satisfies a Łojasiewicz inequality. We prove that the above scheme is Lyapunov stable and that any bounded sequence $(U^n)_n$ which complies with it converges to a critical point of $F$ as $n$ tends to $\infty$. We also obtain convergence rates. We assume that $F$ is semiconvex for some constant $c_F\ge 0$ and that $1/\Delta t

Keywords: proximal method, backward Euler scheme, gradient-like systems, Łojasiewicz inequality.
Mathematics Subject Classification: Primary: 65L20, 65L12; Secondary: 65M12, 65M60.
Citation: Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
References:
[1]

P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions,, SIAM J. Optim., 16 (2005), 531.  doi: 10.1137/040605266.  Google Scholar

[2]

H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features,, Math. Program., 116 (2009), 5.   Google Scholar

[3]

H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and {PDE's},, J. Convex Anal., 15 (2008), 485.   Google Scholar

[4]

T. Bárta, R. Chill and E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system,, submitted., ().  doi: 10.1007/s00605-011-0322-4.  Google Scholar

[5]

I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptot. Anal., 69 (2010), 31.   Google Scholar

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P. Bénilan, M. G. Crandall and A. Pazy, "Bonnes solutions'' d'un problème d'évolution semi-linéaire,, C. R. Acad. Sci. Paris S\'er. I Math., 306 (1988), 527.   Google Scholar

[7]

J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity,, Trans. Amer. Math. Soc., 362 (2010), 3319.   Google Scholar

[8]

H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Publishing Co., (1973).   Google Scholar

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J. V. Chaparova, L. A. Peletier and S. A. Tersian, Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations,, Adv. Differential Equations, 8 (2003), 1237.   Google Scholar

[10]

R. Chill and A. Haraux and M. A. Jendoubi, Applications of the Łojasiewicz-Simon gradient inequality to gradient-like evolution equations,, Anal. Appl. (Singap.), 7 (2009), 351.   Google Scholar

[11]

R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2003), 1017.   Google Scholar

[12]

K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,, Phys. Rev. E, 70 (2004).   Google Scholar

[13]

K. R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing,, Phys. Rev. B, 75 (2007).   Google Scholar

[14]

H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505.   Google Scholar

[15]

P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, Phys. Rev. E, 79 (2009).  doi: 10.1103/PhysRevE.79.051110.  Google Scholar

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M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system,, Commun. Pure Appl. Anal., 5 (2006), 827.   Google Scholar

[17]

M. Grasselli, H. Wu and S. Zheng, Asymptotic behavior of a nonisothermal Ginzburg-Landau model,, Quart. Appl. Math., 66 (2008), 743.   Google Scholar

[18]

A. Haraux, "Syst\`emes dynamiques dissipatifs et applications,'', Masson, (1991).   Google Scholar

[19]

A. Haraux, Slow and fast decay of solutions to some second order evolution equations,, J. Anal. Math., 95 (2005), 297.  doi: 10.1007/BF02791505.  Google Scholar

[20]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities,, J. Differential Equations, 144 (1998), 313.  doi: 10.1006/jdeq.1997.3393.  Google Scholar

[21]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, Calc. Var. Partial Differential Equations, 9 (1999), 95.  doi: 10.1007/s005260050133.  Google Scholar

[22]

S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006).   Google Scholar

[23]

S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal., 46 (2001), 675.  doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[24]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187.  doi: 10.1006/jfan.1997.3174.  Google Scholar

[25]

M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity,, J. Differential Equations, 144 (1998), 302.  doi: 10.1006/jdeq.1997.3392.  Google Scholar

[26]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, in, (1962).   Google Scholar

[27]

S. Łojasiewicz, "Ensembles semi-analytiques,", I.H.E.S. Notes, (1965).   Google Scholar

[28]

P.-E. Maingé, Asymptotic convergence of an inertial proximal method for unconstrained quasiconvex minimization,, J. Global Optim., 45 (2009), 631.  doi: 10.1007/s10898-008-9388-5.  Google Scholar

[29]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications,, Commun. Pure Appl. Anal., 9 (2010), 685.  doi: 10.3934/cpaa.2010.9.685.  Google Scholar

[30]

M. Polat, Global attractor for a modified Swift-Hohenberg equation,, Comput. Math. Appl., 57 (2009), 62.  doi: 10.1016/j.camwa.2008.09.028.  Google Scholar

[31]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,, Ann. of Math., 118 (1983), 525.  doi: 10.2307/2006981.  Google Scholar

[32]

L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equation in $H^k$ spaces,, Nonlinear Anal., 72 (2010), 183.  doi: 10.1016/j.na.2009.06.103.  Google Scholar

[33]

A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,'', Cambridge University Press, (1996).   Google Scholar

[34]

J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319.  doi: 10.1103/PhysRevA.15.319.  Google Scholar

[35]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[36]

S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities,, Discrete Contin. Dyn. Syst., 11 (2004), 351.  doi: 10.3934/dcds.2004.11.351.  Google Scholar

show all references

References:
[1]

P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions,, SIAM J. Optim., 16 (2005), 531.  doi: 10.1137/040605266.  Google Scholar

[2]

H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features,, Math. Program., 116 (2009), 5.   Google Scholar

[3]

H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and {PDE's},, J. Convex Anal., 15 (2008), 485.   Google Scholar

[4]

T. Bárta, R. Chill and E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system,, submitted., ().  doi: 10.1007/s00605-011-0322-4.  Google Scholar

[5]

I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptot. Anal., 69 (2010), 31.   Google Scholar

[6]

P. Bénilan, M. G. Crandall and A. Pazy, "Bonnes solutions'' d'un problème d'évolution semi-linéaire,, C. R. Acad. Sci. Paris S\'er. I Math., 306 (1988), 527.   Google Scholar

[7]

J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity,, Trans. Amer. Math. Soc., 362 (2010), 3319.   Google Scholar

[8]

H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Publishing Co., (1973).   Google Scholar

[9]

J. V. Chaparova, L. A. Peletier and S. A. Tersian, Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations,, Adv. Differential Equations, 8 (2003), 1237.   Google Scholar

[10]

R. Chill and A. Haraux and M. A. Jendoubi, Applications of the Łojasiewicz-Simon gradient inequality to gradient-like evolution equations,, Anal. Appl. (Singap.), 7 (2009), 351.   Google Scholar

[11]

R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2003), 1017.   Google Scholar

[12]

K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,, Phys. Rev. E, 70 (2004).   Google Scholar

[13]

K. R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing,, Phys. Rev. B, 75 (2007).   Google Scholar

[14]

H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505.   Google Scholar

[15]

P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, Phys. Rev. E, 79 (2009).  doi: 10.1103/PhysRevE.79.051110.  Google Scholar

[16]

M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system,, Commun. Pure Appl. Anal., 5 (2006), 827.   Google Scholar

[17]

M. Grasselli, H. Wu and S. Zheng, Asymptotic behavior of a nonisothermal Ginzburg-Landau model,, Quart. Appl. Math., 66 (2008), 743.   Google Scholar

[18]

A. Haraux, "Syst\`emes dynamiques dissipatifs et applications,'', Masson, (1991).   Google Scholar

[19]

A. Haraux, Slow and fast decay of solutions to some second order evolution equations,, J. Anal. Math., 95 (2005), 297.  doi: 10.1007/BF02791505.  Google Scholar

[20]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities,, J. Differential Equations, 144 (1998), 313.  doi: 10.1006/jdeq.1997.3393.  Google Scholar

[21]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, Calc. Var. Partial Differential Equations, 9 (1999), 95.  doi: 10.1007/s005260050133.  Google Scholar

[22]

S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006).   Google Scholar

[23]

S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal., 46 (2001), 675.  doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[24]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187.  doi: 10.1006/jfan.1997.3174.  Google Scholar

[25]

M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity,, J. Differential Equations, 144 (1998), 302.  doi: 10.1006/jdeq.1997.3392.  Google Scholar

[26]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, in, (1962).   Google Scholar

[27]

S. Łojasiewicz, "Ensembles semi-analytiques,", I.H.E.S. Notes, (1965).   Google Scholar

[28]

P.-E. Maingé, Asymptotic convergence of an inertial proximal method for unconstrained quasiconvex minimization,, J. Global Optim., 45 (2009), 631.  doi: 10.1007/s10898-008-9388-5.  Google Scholar

[29]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications,, Commun. Pure Appl. Anal., 9 (2010), 685.  doi: 10.3934/cpaa.2010.9.685.  Google Scholar

[30]

M. Polat, Global attractor for a modified Swift-Hohenberg equation,, Comput. Math. Appl., 57 (2009), 62.  doi: 10.1016/j.camwa.2008.09.028.  Google Scholar

[31]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,, Ann. of Math., 118 (1983), 525.  doi: 10.2307/2006981.  Google Scholar

[32]

L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equation in $H^k$ spaces,, Nonlinear Anal., 72 (2010), 183.  doi: 10.1016/j.na.2009.06.103.  Google Scholar

[33]

A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,'', Cambridge University Press, (1996).   Google Scholar

[34]

J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319.  doi: 10.1103/PhysRevA.15.319.  Google Scholar

[35]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[36]

S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities,, Discrete Contin. Dyn. Syst., 11 (2004), 351.  doi: 10.3934/dcds.2004.11.351.  Google Scholar

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