• Previous Article
    On some spectral problems arising in dynamic populations
  • CPAA Home
  • This Issue
  • Next Article
    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 & 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. Google Scholar

[2]

I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptotic Analysis, 69 (2010), 31-44. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2000), 1017-1039. Google Scholar

[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., ().   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

S. Łojasiewicz, "Ensembles semi-analytiques," Preprint, I.H.E.S., Bures-sur-Yvette, 1965. Google Scholar

[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. Google Scholar

[21]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702. Google Scholar

[22]

J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'' Springer Verlag, New York, Heidelberg, Berlin, 1982. Google Scholar

[23]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571. 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), 531-547. Google Scholar

[2]

I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptotic Analysis, 69 (2010), 31-44. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2000), 1017-1039. Google Scholar

[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., ().   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

S. Łojasiewicz, "Ensembles semi-analytiques," Preprint, I.H.E.S., Bures-sur-Yvette, 1965. Google Scholar

[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. Google Scholar

[21]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702. Google Scholar

[22]

J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'' Springer Verlag, New York, Heidelberg, Berlin, 1982. Google Scholar

[23]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571. Google Scholar

[1]

Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014

[2]

Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations & Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064

[3]

Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure & Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49

[4]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

[5]

Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525

[6]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002

[7]

Santiago Cano-Casanova. Decay rate at infinity of the positive solutions of a generalized class of $T$homas-Fermi equations. Conference Publications, 2011, 2011 (Special) : 240-249. doi: 10.3934/proc.2011.2011.240

[8]

Tomáš Bárta. Exact rate of decay for solutions to damped second order ODE's with a degenerate potential. Evolution Equations & Control Theory, 2018, 7 (4) : 531-543. doi: 10.3934/eect.2018025

[9]

Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021041

[10]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210

[11]

Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031

[12]

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

[13]

Eric Cancès, Claude Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 449-470. doi: 10.3934/dcdsb.2006.6.449

[14]

Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835

[15]

Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837

[16]

Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic & Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007

[17]

Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021

[18]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851

[19]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021

[20]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (255)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]