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Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems
November  2012, 11(6): 2417-2427. doi: 10.3934/cpaa.2012.11.2417

## Some applications of the Łojasiewicz gradient inequality

 1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received  October 2010 Revised  November 2010 Published  April 2012

In the present survey paper, basic convergence results for gradient-like systems relying on the Łojasiewicz gradient inequality are recalled in a self-contained way. A uniform version of the gradient inequality is used to get directly convergence and the rate of convergence in one step and a new technical trick, consisting in the evaluation of the integral of the velocity norm from $t$ to $2t$ is introduced. A short idea of the state of the art without technical details is also given.
Citation: Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417
##### 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), 531-547. [2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptotic Analysis, 69 (2010), 31-44. [3] I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, J. Dynam. Diff. Equa., 23 (2011), 315-332. [4] I. Ben Hassen and A. Haraux, Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy, J. Funct. Anal., 260 (2011), 2933-2963. [5] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Diff. Equa., 20 (2008), 643-652. [6] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity, J. Evol. Equ., 9 (2009), 405-418. [7] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2000), 1017-1039. [8] M. Grasselli and M. Pierre, Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems,, to appear., (). [9] 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. [10] 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. [11] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Ana., 26 (2001), 21-36. [12] A. Haraux and M. A. Jendoubi, The Łojasiewicz gradient inequality in the infinite dimensional Hilbert space framework, J. Funct. Anal., 260 (2011), 2826-2842. [13] A. Haraux, M. A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ., 3 (2003), 463-484. [14] S.-Z. Huang, "Gradient Inequalities,'' American Mathematical Society, Providence, RI, 2006. [15] S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. Ser. A, 46 (2001), 675-698. [16] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. [17] 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. [18] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris (1962), Editions du C.N.R.S., Paris, 1963, 87-89. [19] S. Łojasiewicz, "Ensembles semi-analytiques," Preprint, I.H.E.S., Bures-sur-Yvette, 1965. [20] S. Łojasiewicz, Sur les trajectoires du gradient d'une fonction analytique, Geometry seminars, 1982-1983 (Bologna, 1982-1983 Univ. Stud. Bologna, Bologna, 1984), 115-117. [21] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702. [22] J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'' Springer Verlag, New York, Heidelberg, Berlin, 1982. [23] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571.

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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), 531-547. [2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptotic Analysis, 69 (2010), 31-44. [3] I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, J. Dynam. Diff. Equa., 23 (2011), 315-332. [4] I. Ben Hassen and A. Haraux, Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy, J. Funct. Anal., 260 (2011), 2933-2963. [5] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Diff. Equa., 20 (2008), 643-652. [6] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity, J. Evol. Equ., 9 (2009), 405-418. [7] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2000), 1017-1039. [8] M. Grasselli and M. Pierre, Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems,, to appear., (). [9] 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. [10] 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. [11] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Ana., 26 (2001), 21-36. [12] A. Haraux and M. A. Jendoubi, The Łojasiewicz gradient inequality in the infinite dimensional Hilbert space framework, J. Funct. Anal., 260 (2011), 2826-2842. [13] A. Haraux, M. A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ., 3 (2003), 463-484. [14] S.-Z. Huang, "Gradient Inequalities,'' American Mathematical Society, Providence, RI, 2006. [15] S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. Ser. A, 46 (2001), 675-698. [16] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. [17] 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. [18] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris (1962), Editions du C.N.R.S., Paris, 1963, 87-89. [19] S. Łojasiewicz, "Ensembles semi-analytiques," Preprint, I.H.E.S., Bures-sur-Yvette, 1965. [20] S. Łojasiewicz, Sur les trajectoires du gradient d'une fonction analytique, Geometry seminars, 1982-1983 (Bologna, 1982-1983 Univ. Stud. Bologna, Bologna, 1984), 115-117. [21] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702. [22] J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'' Springer Verlag, New York, Heidelberg, Berlin, 1982. [23] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571.
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