Gamma-convergence of gradient flows on Hilbert and metric spaces and applications

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December 2011, 31(4): 1427-1451. doi: 10.3934/dcds.2011.31.1427

Gamma-convergence of gradient flows on Hilbert and metric spaces and applications

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

Received May 2009 Revised July 2010 Published September 2011

Abstract
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We are concerned with $\Gamma$-convergence of gradient flows, which is a notion meant to ensure that if a family of energy functionals depending of a parameter $\Gamma$-converges, then the solutions to the associated gradient flows converge as well. In this paper we present both a review of the abstract "theory" and of the applications it has had, and a generalization of the scheme to metric spaces which has not appeared elsewhere. We also mention open problems and perspectives.

Keywords: Allen-Cahn equation, Ginzburg-Landau equation, Gamma-convergence, Mullins-Sekerka evolution., Cahn-Hilliard equation, Gradient flow.

Mathematics Subject Classification: Primary: 35K15, 35K99; Secondary: 35Q57, 35Q5.

Citation: Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427

References:

[1]	N. Alikakos, P. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rational Mech. Anal., 128 (1994), 165. doi: 10.1007/BF00375025.
[2]	L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci, 19 (1995), 191.
[3]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures,", Second edition, (2008).
[4]	L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Ann. Institut Henri Poincaré Anal. Non Linéaire, 28 (2011), 217.
[5]	L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495. doi: 10.1002/cpa.20223.
[6]	F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices,", Progress in Nonlinear Differential Equations and their Applications, 13 (1994).
[7]	F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature,, Ann. of Math. (2), 163 (2006), 37. doi: 10.4007/annals.2006.163.37.
[8]	F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics,, Duke Math. J., 130 (2005), 523. doi: 10.1215/S0012-7094-05-13034-4.
[9]	F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices,, Arch. Ration. Mech. Anal., 183 (2007), 315. doi: 10.1007/s00205-006-0018-4.
[10]	F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices,, omm. Math. Phys., 272 (2007), 229. doi: 10.1007/s00220-007-0206-6.
[11]	L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics,, J. Differential Equations, 90 (1991), 211.
[12]	S. J. Chapman, J. Rubinstein and M. Schatzman, A mean-field model of superconducting vortices,, European J. Appl. Math., 7 (1996), 97. doi: 10.1017/S0956792500002242.
[13]	X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116. doi: 10.1016/0022-0396(92)90146-E.
[14]	E. De Giorgi, New problems in $\Gamma$-convergence and $G$-convergence,, in, (1980), 183.
[15]	E. De Giorgi, "New Problems on Minimizing Movements. Boundary Value Problems for Partial Differential Equations and Applications,", 81-98, 29 (1993), 81.
[16]	E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve,, Att Accad Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180.
[17]	M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity,, Nonlinear Anal., 9 (1985), 1401. doi: 10.1016/0362-546X(85)90098-7.
[18]	P. de Mottoni and M. Schatzman, Development of interfaces in $R^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207.
[19]	W. E, W. Ren and E. Vanden-Eijnden, Minimum action method for the study of rare events,, Comm. Pure Appl. Math., 57 (2004), 637.
[20]	L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature,, Comm. Pure Appl. Math, 45 (1992), 1097. doi: 10.1002/cpa.3160450903.
[21]	J. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory,, Calc. Var. Partial Differential Equations, 10 (2000), 49.
[22]	T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature,, J. Differential Geom, 38 (1993), 417.
[23]	R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation,, Calc. Var. Partial Differential Equations, 9 (1999), 1.
[24]	R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices,, Arch. Rational Mech. Anal., 142 (1998), 99. doi: 10.1007/s002050050085.
[25]	R. L. Jerrard and P. Sternberg, Critical points via Gamma-convergence: General theory and applications,, Jour. Eur. Math. Soc., 11 (2009), 705. doi: 10.4171/JEMS/164.
[26]	H. Jian, A relation between $\Gamma$-convergence of functionals and their associated gradient flows,, Sci. China Ser. A, 42 (1999), 133. doi: 10.1007/BF02876564.
[27]	R. V. Kohn, F. Otto, M. G. Reznikoff and E. Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 60 (2007), 393. doi: 10.1002/cpa.20144.
[28]	R. V. Kohn, M. G. Reznikoff and Y. Tonegawa, Sharp-interface limit of the Allen-Cahn action functional in one space dimension,, Calc. Var. Partial Differential Equations, 25 (2006), 503.
[29]	M. Kurzke, The gradient flow motion of boundary vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91.
[30]	N. Le, A Gamma-Convergence approach to the Cahn-Hilliard equation,, Calc. Var. Partial Differential Equations, 32 (2008), 499.
[31]	N. Le, On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law,, SIAM. J. Math. Analysis., 42 (2010), 1602. doi: 10.1137/090768643.
[32]	F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices,, Comm. Pure Appl. Math., 49 (1996), 323.
[33]	A. Marino, C. Saccon and M. Tosques, Curves of maximal slope and parabolic variational inequalities on nonconvex constraints,, Ann Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 281.
[34]	A. Mielke, Weak convergence methods for Hamiltonian multiscale problems,, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 53. doi: 10.3934/dcds.2008.20.53.
[35]	A. Mielke, T. Roubi\vcek and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. PDE, 31 (2008), 387. doi: 10.1007/s00526-007-0119-4.
[36]	L. Modica and S. Mortola, Il limite nella $\Gamma $-convergenza di una famiglia di funzionali ellittici,, Boll. Un. Mat. Ital. A (5), 14 (1977), 526.
[37]	L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions,, Interfaces Free Bound., 10 (2008), 45. doi: 10.4171/IFB/179.
[38]	L. Mugnai and M. Röger, Convergence of the perturbed Allen-Cahn equations to forced mean curvature flow,, preprint., ().
[39]	M. Novaga and E. Valdinocci, Closed curves of prescribed curvature and a pinning effect,, Networks Heterog. Media \textbf{6} (2011), 6 (2011), 77.
[40]	C. Ortner, "Two Variational Techniques for the Approximation of Curves of Maximal Slope,", Technical report NA05/10, (2005).
[41]	C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies,, SIAM J. Math. Anal., 38 (2006), 1214. doi: 10.1137/050643982.
[42]	R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. Roy. Soc. London Ser. A, 422 (1989), 261. doi: 10.1098/rspa.1989.0027.
[43]	M. Röger and R. Schätzle, On a modified conjecture of De Giorgi,, Math. Z., 254 (2006), 675. doi: 10.1007/s00209-006-0002-6.
[44]	E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math, 57 (2004), 1627. doi: 10.1002/cpa.20046.
[45]	E. Sandier and S. Serfaty, Limiting vorticities for the Ginzburg-Landau equations,, Duke Math J., 117 (2003), 403. doi: 10.1215/S0012-7094-03-11732-9.
[46]	E. Sandier and S. Serfaty, A product-estimate for Ginzburg-Landau and corollaries,, J. Func. Anal., 211 (2004), 219. doi: 10.1016/S0022-1236(03)00199-X.
[47]	E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,", Progress in Nonlinear Differential Equations and their Applications, 70 (2007).
[48]	E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity,, Annales Scientifiques de l'ENS (4), 33 (2000), 561.
[49]	N. Sato, A simple proof of convergence of the Allen-Cahn Equation to Brakke's motion by mean curvature,, Indiana Univ. Math. J, 57 (2008), 1743. doi: 10.1512/iumj.2008.57.3283.
[50]	, R. Schätzle,, private communication., ().
[51]	S. Serfaty, Stability in 2D Ginzburg-Landau passes to the limit,, Indiana Univ. Math. J., 54 (2005), 199. doi: 10.1512/iumj.2005.54.2497.
[52]	S. Serfaty, Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow. Part I: Study of the perturbed Ginzburg-Landau equation,, Journal Eur. Math Society, 9 (2007), 177. doi: 10.4171/JEMS/84.
[53]	C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009).
[54]	M. G. Westdickenberg and Y. Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional,, Indiana Univ. Math. J., 56 (2007), 2935. doi: 10.1512/iumj.2007.56.3182.

show all references

References:

[1]	N. Alikakos, P. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rational Mech. Anal., 128 (1994), 165. doi: 10.1007/BF00375025.
[2]	L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci, 19 (1995), 191.
[3]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures,", Second edition, (2008).
[4]	L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Ann. Institut Henri Poincaré Anal. Non Linéaire, 28 (2011), 217.
[5]	L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495. doi: 10.1002/cpa.20223.
[6]	F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices,", Progress in Nonlinear Differential Equations and their Applications, 13 (1994).
[7]	F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature,, Ann. of Math. (2), 163 (2006), 37. doi: 10.4007/annals.2006.163.37.
[8]	F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics,, Duke Math. J., 130 (2005), 523. doi: 10.1215/S0012-7094-05-13034-4.
[9]	F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices,, Arch. Ration. Mech. Anal., 183 (2007), 315. doi: 10.1007/s00205-006-0018-4.
[10]	F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices,, omm. Math. Phys., 272 (2007), 229. doi: 10.1007/s00220-007-0206-6.
[11]	L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics,, J. Differential Equations, 90 (1991), 211.
[12]	S. J. Chapman, J. Rubinstein and M. Schatzman, A mean-field model of superconducting vortices,, European J. Appl. Math., 7 (1996), 97. doi: 10.1017/S0956792500002242.
[13]	X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116. doi: 10.1016/0022-0396(92)90146-E.
[14]	E. De Giorgi, New problems in $\Gamma$-convergence and $G$-convergence,, in, (1980), 183.
[15]	E. De Giorgi, "New Problems on Minimizing Movements. Boundary Value Problems for Partial Differential Equations and Applications,", 81-98, 29 (1993), 81.
[16]	E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve,, Att Accad Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180.
[17]	M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity,, Nonlinear Anal., 9 (1985), 1401. doi: 10.1016/0362-546X(85)90098-7.
[18]	P. de Mottoni and M. Schatzman, Development of interfaces in $R^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207.
[19]	W. E, W. Ren and E. Vanden-Eijnden, Minimum action method for the study of rare events,, Comm. Pure Appl. Math., 57 (2004), 637.
[20]	L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature,, Comm. Pure Appl. Math, 45 (1992), 1097. doi: 10.1002/cpa.3160450903.
[21]	J. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory,, Calc. Var. Partial Differential Equations, 10 (2000), 49.
[22]	T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature,, J. Differential Geom, 38 (1993), 417.
[23]	R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation,, Calc. Var. Partial Differential Equations, 9 (1999), 1.
[24]	R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices,, Arch. Rational Mech. Anal., 142 (1998), 99. doi: 10.1007/s002050050085.
[25]	R. L. Jerrard and P. Sternberg, Critical points via Gamma-convergence: General theory and applications,, Jour. Eur. Math. Soc., 11 (2009), 705. doi: 10.4171/JEMS/164.
[26]	H. Jian, A relation between $\Gamma$-convergence of functionals and their associated gradient flows,, Sci. China Ser. A, 42 (1999), 133. doi: 10.1007/BF02876564.
[27]	R. V. Kohn, F. Otto, M. G. Reznikoff and E. Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 60 (2007), 393. doi: 10.1002/cpa.20144.
[28]	R. V. Kohn, M. G. Reznikoff and Y. Tonegawa, Sharp-interface limit of the Allen-Cahn action functional in one space dimension,, Calc. Var. Partial Differential Equations, 25 (2006), 503.
[29]	M. Kurzke, The gradient flow motion of boundary vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91.
[30]	N. Le, A Gamma-Convergence approach to the Cahn-Hilliard equation,, Calc. Var. Partial Differential Equations, 32 (2008), 499.
[31]	N. Le, On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law,, SIAM. J. Math. Analysis., 42 (2010), 1602. doi: 10.1137/090768643.
[32]	F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices,, Comm. Pure Appl. Math., 49 (1996), 323.
[33]	A. Marino, C. Saccon and M. Tosques, Curves of maximal slope and parabolic variational inequalities on nonconvex constraints,, Ann Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 281.
[34]	A. Mielke, Weak convergence methods for Hamiltonian multiscale problems,, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 53. doi: 10.3934/dcds.2008.20.53.
[35]	A. Mielke, T. Roubi\vcek and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. PDE, 31 (2008), 387. doi: 10.1007/s00526-007-0119-4.
[36]	L. Modica and S. Mortola, Il limite nella $\Gamma $-convergenza di una famiglia di funzionali ellittici,, Boll. Un. Mat. Ital. A (5), 14 (1977), 526.
[37]	L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions,, Interfaces Free Bound., 10 (2008), 45. doi: 10.4171/IFB/179.
[38]	L. Mugnai and M. Röger, Convergence of the perturbed Allen-Cahn equations to forced mean curvature flow,, preprint., ().
[39]	M. Novaga and E. Valdinocci, Closed curves of prescribed curvature and a pinning effect,, Networks Heterog. Media \textbf{6} (2011), 6 (2011), 77.
[40]	C. Ortner, "Two Variational Techniques for the Approximation of Curves of Maximal Slope,", Technical report NA05/10, (2005).
[41]	C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies,, SIAM J. Math. Anal., 38 (2006), 1214. doi: 10.1137/050643982.
[42]	R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. Roy. Soc. London Ser. A, 422 (1989), 261. doi: 10.1098/rspa.1989.0027.
[43]	M. Röger and R. Schätzle, On a modified conjecture of De Giorgi,, Math. Z., 254 (2006), 675. doi: 10.1007/s00209-006-0002-6.
[44]	E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math, 57 (2004), 1627. doi: 10.1002/cpa.20046.
[45]	E. Sandier and S. Serfaty, Limiting vorticities for the Ginzburg-Landau equations,, Duke Math J., 117 (2003), 403. doi: 10.1215/S0012-7094-03-11732-9.
[46]	E. Sandier and S. Serfaty, A product-estimate for Ginzburg-Landau and corollaries,, J. Func. Anal., 211 (2004), 219. doi: 10.1016/S0022-1236(03)00199-X.
[47]	E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,", Progress in Nonlinear Differential Equations and their Applications, 70 (2007).
[48]	E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity,, Annales Scientifiques de l'ENS (4), 33 (2000), 561.
[49]	N. Sato, A simple proof of convergence of the Allen-Cahn Equation to Brakke's motion by mean curvature,, Indiana Univ. Math. J, 57 (2008), 1743. doi: 10.1512/iumj.2008.57.3283.
[50]	, R. Schätzle,, private communication., ().
[51]	S. Serfaty, Stability in 2D Ginzburg-Landau passes to the limit,, Indiana Univ. Math. J., 54 (2005), 199. doi: 10.1512/iumj.2005.54.2497.
[52]	S. Serfaty, Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow. Part I: Study of the perturbed Ginzburg-Landau equation,, Journal Eur. Math Society, 9 (2007), 177. doi: 10.4171/JEMS/84.
[53]	C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009).
[54]	M. G. Westdickenberg and Y. Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional,, Indiana Univ. Math. J., 56 (2007), 2935. doi: 10.1512/iumj.2007.56.3182.

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