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Global attractor for damped forced nonlinear logarithmic Schrödinger equations
On the convergence to equilibria of a sequence defined by an implicit scheme
1. | Laboratoire M2N, EA7340, CNAM, 292 rue Saint-Martin, 75003, Paris, France |
2. | Université de Carthage, Institut Préparatoire aux Etudes Scientifiques et Techniques, B.P. 51 2070 La Marsa, Tunisie |
3. | Laboratoire équations aux dérivées partielles, Faculté des sciences de Tunis, Université Tunis El Manar, Campus universitaire El-Manar, 2092 El-Manar, Tunisie |
We present numerical implicit schemes based on a geometric approach of the study of the convergence of solutions of gradient-like systems given in [
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.
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), 5-16.
doi: 10.1007/s10107-007-0133-5. |
[3] |
T. Barta, R. Chill and E. Fašangová,
Every ordinary differential equation with a strict Lyapunov function is a gradient system, Monatsh. Math., 166 (2012), 57-72.
doi: 10.1007/s00605-011-0322-4. |
[4] |
J. Bolte, A. Daniilidis, O. Ley and L. Mazet,
Characterizations of Lojasiewicz inequalities and applications, Trans. Amer. Math. Soc, 362 (6) (2010), 3319-3363.
doi: 10.1090/S0002-9947-09-05048-X. |
[5] |
R. Chill, A. Haraux and M. A. Jendoubi,
Applications of the Lojasiewicz-Simon gradient inequality to gradient-like evolution equations, Anal. Appl., 7 (2009), 351-372.
doi: 10.1142/S0219530509001438. |
[6] |
A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems. Classical Methods and Recent Advances, SpringerBriefs in Mathematics. Cham : Springer. 2015
doi: 10.1007/978-3-319-23407-6. |
[7] |
J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, AMS, 2006. Google Scholar |
[8] |
M. A. Jendoubi, Convergence des solutions globales et bornées de quelques problèmes d'évolution avec nonlinéarité analytique, in Progress in Partial Differential Equations. Papers from the 3rd European conference on elliptic and parabolic problems, Pont-à-Mousson, France, June 1997. Vol. 1. (eds Amann, H. (ed.) et al.) Harlow: Longman. Pitman Res. Notes Math. Ser., 383 (1998), 181–190. |
[9] |
W. Klingenberg, Riemannian Geometry, De Gruyter Studies in Mathematics, 1, Berlin: Walter de Gruyter & Co. 1982. |
[10] |
S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S, Bures-sur-Yvette, 1965. Google Scholar |
[11] |
S. Lojasiewicz, Une proprièté topologique des sous ensembles analytiques réels, in Les Équations aux Dérivées Partielles, Colloques internationaux du C.N.R.S, 117. 1963. |
[12] |
B. Merlet and M. Pierre,
Convergence to equilibrium for the backward Euler scheme and applications, Comm. Pure and Appl. Anal., 9 (2010), 665-702.
doi: 10.3934/cpaa.2010.9.685. |
[13] |
B. Merlet and T. H. Nguyen, Convergence to equilibrium for the backward euler scheme and applicationsconvergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds, Differential Integral Equations 26 (2013), 571–602. https://projecteuclid.org/euclid.die/1363266079 Google Scholar |
[14] |
J. Nash,
The imbedding problem for Riemannian manifolds, Annals of Maths, 63 (1956), 20-63.
doi: 10.2307/1969989. |
[15] |
J. H. C. Whitehead, Convex regions in the geometry of paths, Quart. J. Math., Oxford 3 (1932), 33–42, .
doi: 10.1093/qmath/os-3.1.33. |
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.
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), 5-16.
doi: 10.1007/s10107-007-0133-5. |
[3] |
T. Barta, R. Chill and E. Fašangová,
Every ordinary differential equation with a strict Lyapunov function is a gradient system, Monatsh. Math., 166 (2012), 57-72.
doi: 10.1007/s00605-011-0322-4. |
[4] |
J. Bolte, A. Daniilidis, O. Ley and L. Mazet,
Characterizations of Lojasiewicz inequalities and applications, Trans. Amer. Math. Soc, 362 (6) (2010), 3319-3363.
doi: 10.1090/S0002-9947-09-05048-X. |
[5] |
R. Chill, A. Haraux and M. A. Jendoubi,
Applications of the Lojasiewicz-Simon gradient inequality to gradient-like evolution equations, Anal. Appl., 7 (2009), 351-372.
doi: 10.1142/S0219530509001438. |
[6] |
A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems. Classical Methods and Recent Advances, SpringerBriefs in Mathematics. Cham : Springer. 2015
doi: 10.1007/978-3-319-23407-6. |
[7] |
J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, AMS, 2006. Google Scholar |
[8] |
M. A. Jendoubi, Convergence des solutions globales et bornées de quelques problèmes d'évolution avec nonlinéarité analytique, in Progress in Partial Differential Equations. Papers from the 3rd European conference on elliptic and parabolic problems, Pont-à-Mousson, France, June 1997. Vol. 1. (eds Amann, H. (ed.) et al.) Harlow: Longman. Pitman Res. Notes Math. Ser., 383 (1998), 181–190. |
[9] |
W. Klingenberg, Riemannian Geometry, De Gruyter Studies in Mathematics, 1, Berlin: Walter de Gruyter & Co. 1982. |
[10] |
S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S, Bures-sur-Yvette, 1965. Google Scholar |
[11] |
S. Lojasiewicz, Une proprièté topologique des sous ensembles analytiques réels, in Les Équations aux Dérivées Partielles, Colloques internationaux du C.N.R.S, 117. 1963. |
[12] |
B. Merlet and M. Pierre,
Convergence to equilibrium for the backward Euler scheme and applications, Comm. Pure and Appl. Anal., 9 (2010), 665-702.
doi: 10.3934/cpaa.2010.9.685. |
[13] |
B. Merlet and T. H. Nguyen, Convergence to equilibrium for the backward euler scheme and applicationsconvergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds, Differential Integral Equations 26 (2013), 571–602. https://projecteuclid.org/euclid.die/1363266079 Google Scholar |
[14] |
J. Nash,
The imbedding problem for Riemannian manifolds, Annals of Maths, 63 (1956), 20-63.
doi: 10.2307/1969989. |
[15] |
J. H. C. Whitehead, Convex regions in the geometry of paths, Quart. J. Math., Oxford 3 (1932), 33–42, .
doi: 10.1093/qmath/os-3.1.33. |
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