Some applications of the Łojasiewicz gradient inequality
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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,
Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano
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
P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16 (2005), 531-547 (electronic). doi: 10.1137/040605266. Google Scholar
H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), 5-16. Google Scholar
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), 485-506. Google Scholar
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
I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptot. Anal., 69 (2010), 31-44. Google Scholar
P. Bénilan, M. G. Crandall and A. Pazy, "Bonnes solutions'' d'un problème d'évolution semi-linéaire, C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 527-530. Google Scholar
J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity, Trans. Amer. Math. Soc., 362 (2010), 3319-3363. Google Scholar
H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'' North-Holland Publishing Co., Amsterdam, 1973. Google Scholar
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), 1237-1258. Google Scholar
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), 351-372. Google Scholar
R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2003), 1017-1039. Google Scholar
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), 505-528. Google Scholar
P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110 (11 p.). doi: 10.1103/PhysRevE.79.051110. Google Scholar
M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal., 5 (2006), 827-838. Google Scholar
M. Grasselli, H. Wu and S. Zheng, Asymptotic behavior of a nonisothermal Ginzburg-Landau model, Quart. Appl. Math., 66 (2008), 743-770. Google Scholar
A. Haraux, "Syst\`emes dynamiques dissipatifs et applications,'' Masson, Paris, 1991. Google Scholar
A. Haraux, Slow and fast decay of solutions to some second order evolution equations, J. Anal. Math., 95 (2005), 297-321. doi: 10.1007/BF02791505. Google Scholar
A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320. doi: 10.1006/jdeq.1997.3393. Google Scholar
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), 95-124. doi: 10.1007/s005260050133. Google Scholar
S.-Z. Huang, "Gradient Inequalities,'' American Mathematical Society, Providence, RI, 2006. Google Scholar
S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46 (2001), 675-698. doi: 10.1016/S0362-546X(00)00145-0. Google Scholar
M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174. Google Scholar
M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144 (1998), 302-312. doi: 10.1006/jdeq.1997.3392. Google Scholar
S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in "Les Équations aux Dérivées Partielles (Paris, 1962)" Google Scholar
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), 631-644. doi: 10.1007/s10898-008-9388-5. Google Scholar
B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702. doi: 10.3934/cpaa.2010.9.685. Google Scholar
M. Polat, Global attractor for a modified Swift-Hohenberg equation, Comput. Math. Appl., 57 (2009), 62-66. doi: 10.1016/j.camwa.2008.09.028. Google Scholar
L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571. doi: 10.2307/2006981. Google Scholar
L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equation in $H^k$ spaces, Nonlinear Anal., 72 (2010), 183-191. doi: 10.1016/j.na.2009.06.103. Google Scholar
A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,'' Cambridge University Press, Cambridge, 1996. Google Scholar
J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-329. doi: 10.1103/PhysRevA.15.319. Google Scholar
S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934. doi: 10.3934/cpaa.2004.3.921. Google Scholar
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
Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417
Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 947-973. doi: 10.3934/dcds.2000.6.947
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Maurizio Grasselli Morgan Pierre