×

zbMATH — the first resource for mathematics

Kurdyka-Łojasiewicz-Simon inequality for gradient flows in metric spaces. (English) Zbl 1425.49023
Summary: This paper is dedicated to providing new tools and methods for studying the trend to equilibrium of gradient flows in metric spaces \((\mathfrak{M},d)\) in the entropy and metric sense, to establish decay rates and finite time of extinction, and to characterize Lyapunov stable equilibrium points. More precisely, our main results are as follows:
\(\bullet\)
Introduction of a gradient inequality in the metric space framework, which in the Euclidean space \({\mathbb{R}}^N\) was obtained by S. Łojasiewicz [in: Equ. Derivees Partielles, Paris 1962. Colloques internat. Centre Nat. Rech. Sci. 117, 87–89 (1963; Zbl 0234.57007)], later improved by K. Kurdyka [Ann. Inst. Fourier 48, No. 3, 769–783 (1998; Zbl 0934.32009)], and generalized to the Hilbert space framework by L. Simon [Ann. Math. (2) 118, 525–571 (1983; Zbl 0549.35071)].
\(\bullet\)
Obtainment of the trend to equilibrium in the entropy and metric sense of gradient flows generated by a function \({\mathcal{E}} : \mathfrak{M}\to (-\infty ,+\infty ]\) satisfying a Kurdyka-Łojasiewicz-Simon inequality in a neighborhood of an equilibrium point of \({\mathcal{E}}\). Sufficient conditions are given implying decay rates and finite time of extinction of gradient flows.
\(\bullet\)
Construction of a talweg curve in \(\mathfrak{M}\) with an optimal growth function yielding the validity of a Kurdyka-Łojasiewicz-Simon inequality. Characterization of Lyapunov stable equilibrium points of \({\mathcal{E}}\) satisfying a Kurdyka-Łojasiewicz-Simon inequality near such points.
\(\bullet\)
Characterization of the entropy-entropy production inequality with the Kurdyka-Łojasiewicz-Simon inequality.
As an application of these results, the following properties are established.
\(\bullet\)
New upper bounds on the extinction time of gradient flows associated with the total variational flow.
\(\bullet\)
If the metric space \(\mathfrak{M}\) is the \(p\)-Wasserstein space \(\mathcal{P}_p({\mathbb{R}}^N), 1<p<\infty\), then new HWI-, Talagrand, and logarithmic Sobolev inequalities are obtained for functions \({\mathcal{E}}\) associated with nonlinear diffusion problems modeling drift, potential, and interaction phenomena. It is shown that these inequalities are equivalent to the Kurdyka-Łojasiewicz-Simon inequality, and hence they imply a trend to equilibrium of the gradient flows of \({\mathcal{E}}\) with decay rates or arrival in finite time.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49J52 Nonsmooth analysis
39B62 Functional inequalities, including subadditivity, convexity, etc.
35K90 Abstract parabolic equations
58J35 Heat and other parabolic equation methods for PDEs on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Absil, P.-A.; Kurdyka, K., On the stable equilibrium points of gradient systems, Systems Control Lett., 55, 7, 573-577 (2006) · Zbl 1129.34320
[2] Agueh, M.; Ghoussoub, N.; Kang, X., Geometric inequalities via a general comparison principle for interacting gases, Geom. Funct. Anal., 14, 1, 215-244 (2004) · Zbl 1122.82022
[3] Agueh, Martial, Asymptotic behavior for doubly degenerate parabolic equations, C. R. Math. Acad. Sci. Paris, 337, 5, 331-336 (2003) · Zbl 1029.35144
[4] Agueh, Martial, Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory, Adv. Differential Equations, 10, 3, 309-360 (2005) · Zbl 1103.35051
[5] Ambrosio, Luigi, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19, 191-246 (1995) · Zbl 0957.49029
[6] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, xviii+434 pp. (2000), The Clarendon Press, Oxford University Press, New York · Zbl 0957.49001
[7] Ambrosio, Luigi; Gigli, Nicola; Savar\'{e}, Giuseppe, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Z\`“{u}rich, x+334 pp. (2008), Birkh\'”{a}user Verlag, Basel · Zbl 1145.35001
[8] Ambrosio, Luigi; Gigli, Nicola; Savar\'{e}, Giuseppe, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195, 2, 289-391 (2014) · Zbl 1312.53056
[9] Andreu, F.; Ballester, C.; Caselles, V.; Maz\'{o}n, J. M., The Dirichlet problem for the total variation flow, J. Funct. Anal., 180, 2, 347-403 (2001) · Zbl 0973.35109
[10] Andreu, F.; Caselles, V.; D\'{i}az, J. I.; Maz\'{o}n, J. M., Some qualitative properties for the total variation flow, J. Funct. Anal., 188, 2, 516-547 (2002) · Zbl 1042.35018
[11] Andreu-Vaillo, Fuensanta; Caselles, Vicent; Maz\'{o}n, Jos\'{e} M., Parabolic quasilinear equations minimizing linear growth functionals, Progress in Mathematics 223, xiv+340 pp. (2004), Birkh\"{a}user Verlag, Basel · Zbl 1053.35002
[12] ACDDJLMTV A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. J\"ungel, C. Lederman, P. A. Markowich, G. Toscani, and C. Villani, Entropies and equilibria of many-particle systems: An essay on recent research, Monatsh. Math. 142 (2004), 35-43. http://dx.doi.org/10.1007/s00605-004-0239-2doi:10.1007/s00605-004-0239-2 · Zbl 1063.35109
[13] Baillon, J.-B., Un exemple concernant le comportement asymptotique de la solution du probl\`eme \(du/dt+\partial\varphi(u)\ni0\), J. Funct. Anal., 28, 3, 369-376 (1978) · Zbl 0386.47041
[14] MR889476 D. Bakry and M. \'Emery, Diffusions hypercontractives, S\'eminaire de probabilit\'es, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177-206. http://dx.doi.org/10.1007/BFb0075847doi:10.1007/BFb0075847
[15] Blanchet, Adrien; Bolte, J\'{e}r\^ome, A family of functional inequalities: \L ojasiewicz inequalities and displacement convex functions, J. Funct. Anal., 275, 7, 1650-1673 (2018) · Zbl 1403.39023
[16] MR2274510 J. Bolte, A. Daniilidis, and A. Lewis, The ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim. 17 (2006), 1205-1223. http://dx.doi.org/10.1137/050644641doi:10.1137/050644641 · Zbl 1129.26012
[17] Bolte, J\'{e}r\^ome; Daniilidis, Aris; Ley, Olivier; Mazet, Laurent, Characterizations of \L ojasiewicz inequalities: Subgradient flows, talweg, convexity, Trans. Amer. Math. Soc., 362, 6, 3319-3363 (2010) · Zbl 1202.26026
[18] Bonforte, Matteo; Figalli, Alessio, Total variation flow and sign fast diffusion in one dimension, J. Differential Equations, 252, 8, 4455-4480 (2012) · Zbl 1242.35049
[19] Brenier1 Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417. http://dx.doi.org/10.1002/cpa.3160440402doi:10.1002/cpa.3160440402 · Zbl 0738.46011
[20] Br\'{e}zis, H., Op\'{e}rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, vi+183 pp. (1973), North-Holland Publishing Co., Amsterdam\textendash London; American Elsevier Publishing Co., Inc., New York · Zbl 0252.47055
[21] Brezis, Haim, Functional analysis, Sobolev spaces and partial differential equations, Universitext, xiv+599 pp. (2011), Springer, New York · Zbl 1220.46002
[22] CJMTU J. A. Carrillo, A. J\"ungel, P. A. Markowich, G. Toscani, and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001), 1-82. http://dx.doi.org/10.1007/s006050170032doi:10.1007/s006050170032 · Zbl 0984.35027
[23] Carrillo, Jos\'{e} A.; McCann, Robert J.; Villani, C\'{e}dric, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19, 3, 971-1018 (2003) · Zbl 1073.35127
[24] CMVArchive J. A. Carrillo, R. J. McCann, and C. Villani, Contractions in the \(2-W\) asserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal. 179 (2006), 217-263. http://dx.doi.org/10.1007/s00205-005-0386-1doi:10.1007/s00205-005-0386-1 · Zbl 1082.76105
[25] Chill, Ralph, On the \L ojasiewicz-Simon gradient inequality, J. Funct. Anal., 201, 2, 572-601 (2003) · Zbl 1036.26015
[26] chill-fasangova:isem R. Chill and E. Fasangov\'a, Gradient Systems, MATFYZPRESS, Publishing House of the Faculty of Mathematics and Physics, Charles University, Prague, 2010.
[27] Chill, Ralph; Fiorenza, Alberto, Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations, J. Differential Equations, 228, 2, 611-632 (2006) · Zbl 1115.35021
[28] Chill, Ralph; Hauer, Daniel; Kennedy, James, Nonlinear semigroups generated by \(j\)-elliptic functionals, J. Math. Pures Appl. (9), 105, 3, 415-450 (2016) · Zbl 1332.47031
[29] Chill, Ralph; Mildner, Sebastian, The Kurdyka-\L ojasiewicz-Simon inequality and stabilisation in nonsmooth infinite-dimensional gradient systems, Proc. Amer. Math. Soc., 146, 10, 4307-4314 (2018) · Zbl 1397.34102
[30] Cordero-Erausquin, Dario; Gangbo, Wilfrid; Houdr\'{e}, Christian, Inequalities for generalized entropy and optimal transportation. Recent advances in the theory and applications of mass transport, Contemp. Math. 353, 73-94 (2004), Amer. Math. Soc., Providence, RI · Zbl 1135.49026
[31] Daneri, Sara; Savar\'{e}, Giuseppe, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40, 3, 1104-1122 (2008) · Zbl 1166.58011
[32] Daneri, Sara; Savar\'{e}, Giuseppe, Lecture notes on gradient flows and optimal transport. Optimal transportation, London Math. Soc. Lecture Note Ser. 413, 100-144 (2014), Cambridge Univ. Press, Cambridge, England · Zbl 1333.49001
[33] De Giorgi, Ennio, New problems on minimizing movements. Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math. 29, 81-98 (1993), Masson, Paris · Zbl 0851.35052
[34] De Giorgi, Ennio; Marino, Antonio; Tosques, Mario, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68, 3, 180-187 (1980) · Zbl 0465.47041
[35] 2014arXiv1409.1525F P. M. N. Feehan, Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow, arXiv:1409.1525 (2014).
[36] G-Mc W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), 113-161. http://dx.doi.org/10.1007/BF02392620doi:10.1007/BF02392620 · Zbl 0887.49017
[37] Giga, Yoshikazu; Kohn, Robert V., Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst., 30, 2, 509-535 (2011) · Zbl 1223.35211
[38] Haraux, Alain, Syst\`“emes dynamiques dissipatifs et applications, Recherches en Math\'”{e}matiques Appliqu\'{e}es [Research in Applied Mathematics] 17, xii+132 pp. (1991), Masson, Paris · Zbl 0726.58001
[39] Haraux, Alain; Jendoubi, Mohamed Ali, The convergence problem for dissipative autonomous systems, SpringerBriefs in Mathematics, xii+142 pp. (2015), Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao · Zbl 1345.37081
[40] Haraux, Alain; Jendoubi, Mohamed Ali; Kavian, Otared, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ., 3, 3, 463-484 (2003) · Zbl 1036.35035
[41] Huang, Sen-Zhong, Gradient inequalities, Mathematical Surveys and Monographs 126, viii+184 pp. (2006), American Mathematical Society, Providence, RI · Zbl 1132.35002
[42] Jendoubi, Mohamed Ali, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153, 1, 187-202 (1998) · Zbl 0895.35012
[43] JKO R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), 1-17. http://dx.doi.org/10.1137/S0036141096303359doi:10.1137/S0036141096303359 · Zbl 0915.35120
[44] Kurdyka, Krzysztof, On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier (Grenoble), 48, 3, 769-783 (1998) · Zbl 0934.32009
[45] Leoni, Giovanni, A first course in Sobolev spaces, Graduate Studies in Mathematics 105, xvi+607 pp. (2009), American Mathematical Society, Providence, RI · Zbl 1180.46001
[46] \L ojasiewicz, S., Une propri\'{e}t\'{e} topologique des sous-ensembles analytiques r\'{e}els. Les \'{E}quations aux D\'{e}riv\'{e}es Partielles, Paris, 1962, 87-89 (1963), \'{E}ditions du Centre National de la Recherche Scientifique, Paris
[47] \L ojasiewicz, S., Sur les ensembles semi-analytiques. Actes du Congr\`“es International des Math\'”{e}maticiens, Nice, 1970, 237-241 (1971), Gauthier-Villars, Paris
[48] Lott, John; Villani, C\'{e}dric, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169, 3, 903-991 (2009) · Zbl 1178.53038
[49] Otto, F.; Villani, C., Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173, 2, 361-400 (2000) · Zbl 0985.58019
[50] Otto, Felix, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26, 1-2, 101-174 (2001) · Zbl 0984.35089
[51] Palis, Jacob, Jr.; de Melo, Welington, Geometric theory of dynamical systems, xii+198 pp. (1982), Springer-Verlag, New York\textendash Berlin
[52] Rezakhanlou, Fraydoun; Villani, C\'{e}dric, Entropy methods for the Boltzmann equation, Lecture Notes in Mathematics 1916, xii+107 pp. (2008), Springer, Berlin · Zbl 1125.76001
[53] Rossi, Riccarda; Segatti, Antonio; Stefanelli, Ulisse, Global attractors for gradient flows in metric spaces, J. Math. Pures Appl. (9), 95, 2, 205-244 (2011) · Zbl 1215.35036
[54] Santambrogio, Filippo, Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications 87, xxvii+353 pp. (2015), Birkh\"{a}user/Springer, Cham · Zbl 1401.49002
[55] Simon, Leon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118, 3, 525-571 (1983) · Zbl 0549.35071
[56] Sturm, Karl-Theodor, On the geometry of metric measure spaces. I, Acta Math., 196, 1, 65-131 (2006) · Zbl 1105.53035
[57] Sturm, Karl-Theodor, On the geometry of metric measure spaces. II, Acta Math., 196, 1, 133-177 (2006) · Zbl 1106.53032
[58] Talenti, Giorgio, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110, 353-372 (1976) · Zbl 0353.46018
[59] MR1633348 L. van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, England, 1998. http://dx.doi.org/10.1017/CBO9780511525919doi:10.1017/CBO9780511525919 · Zbl 0953.03045
[60] Villani, Cedric, Optimal transportation, dissipative PDE’s and functional inequalities. Optimal transportation and applications, Martina Franca, 2001, Lecture Notes in Math. 1813, 53-89 (2003), Springer, Berlin · Zbl 1039.35147
[61] Villani, C\'{e}dric, Topics in optimal transportation, Graduate Studies in Mathematics 58, xvi+370 pp. (2003), American Mathematical Society, Providence, RI · Zbl 1013.00028
[62] Villani, C\'{e}dric, Trend to equilibrium for dissipative equations, functional inequalities and mass transportation. Recent advances in the theory and applications of mass transport, Contemp. Math. 353, 95-109 (2004), Amer. Math. Soc., Providence, RI · Zbl 1134.35375
[63] Villani, C\'{e}dric, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338, xxii+973 pp. (2009), Springer-Verlag, Berlin · Zbl 1156.53003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.