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Some applications of the Łojasiewicz gradient inequality
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Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems
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 |
$\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
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.
H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features,, Math. Program., 116 (2009), 5.
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.
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.
I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptot. Anal., 69 (2010), 31.
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.
J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity,, Trans. Amer. Math. Soc., 362 (2010), 3319.
H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Publishing Co., (1973).
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.
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.
R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2003), 1017.
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
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
H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505.
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.
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.
M. Grasselli, H. Wu and S. Zheng, Asymptotic behavior of a nonisothermal Ginzburg-Landau model,, Quart. Appl. Math., 66 (2008), 743.
A. Haraux, "Syst\`emes dynamiques dissipatifs et applications,'', Masson, (1991).
A. Haraux, Slow and fast decay of solutions to some second order evolution equations,, J. Anal. Math., 95 (2005), 297.
doi: 10.1007/BF02791505.
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.
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.
S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006).
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.
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.
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.
S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, in, (1962).
S. Łojasiewicz, "Ensembles semi-analytiques,", I.H.E.S. Notes, (1965).
Google Scholar
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.
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.
M. Polat, Global attractor for a modified Swift-Hohenberg equation,, Comput. Math. Appl., 57 (2009), 62.
doi: 10.1016/j.camwa.2008.09.028.
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.
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.
A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,'', Cambridge University Press, (1996).
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.
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.
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. show all references
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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. |
[2] |
H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features,, Math. Program., 116 (2009), 5.
|
[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.
|
[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. |
[5] |
I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptot. Anal., 69 (2010), 31.
|
[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.
|
[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.
|
[8] |
H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Publishing Co., (1973).
|
[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.
|
[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.
|
[11] |
R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2003), 1017.
|
[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.
|
[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. |
[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.
|
[17] |
M. Grasselli, H. Wu and S. Zheng, Asymptotic behavior of a nonisothermal Ginzburg-Landau model,, Quart. Appl. Math., 66 (2008), 743.
|
[18] |
A. Haraux, "Syst\`emes dynamiques dissipatifs et applications,'', Masson, (1991).
|
[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. |
[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. |
[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. |
[22] |
S.-Z. Huang, "Gradient Inequalities,'', American Mathematical Society, (2006).
|
[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. |
[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. |
[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. |
[26] |
S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels,, in, (1962).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,'', Cambridge University Press, (1996).
|
[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. |
[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. |
[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. |
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