• Previous Article
    Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials
  • DCDS-S Home
  • This Issue
  • Next Article
    Effective diffusion in thin structures via generalized gradient systems and EDP-convergence
doi: 10.3934/dcdss.2020324

Perturbed minimizing movements of families of functionals

Department of Mathematics, University of Rome Tor Vergata, via della Ricerca Scientifica, 00133 Rome, Italy

* Corresponding author: Andrea Braides

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  March 2019 Revised  October 2019 Published  April 2020

We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a ($ \Gamma $-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters $ \varepsilon $ and $ \tau $, governing energy and time scales, respectively. We characterize the extreme cases when $ \varepsilon/\tau $ and $ \tau/ \varepsilon $ converges to $ 0 $ sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of $ \varepsilon $ and $ \tau $. We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies.

Citation: Andrea Braides, Antonio Tribuzio. Perturbed minimizing movements of families of functionals. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020324
References:
[1]

R. AlicandroA. Braides and M. Cicalese, Phase and anti-phase boundaries in binary discrete systems: A variational viewpoint, Netw. Heterog. Media, 1 (2006), 85-107.  doi: 10.3934/nhm.2006.1.85.  Google Scholar

[2]

F. Almgren and J. E. Taylor, Flat flow is motion by mean curvature for curves with crystalline energy, J. Differential Geom., 42 (1995), 1-22.  doi: 10.4310/jdg/1214457030.  Google Scholar

[3]

F. AlmgrenJ. L. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020.  Google Scholar

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Birkhäuser, Basel, 2008.  Google Scholar

[5]

N. AnsiniA. Braides and J. Zimmer, Minimizing movements for oscillating energies: The critical regime, Proc. Royal Soc. Edin. A, 149 (2019), 719-737.  doi: 10.1017/prm.2018.46.  Google Scholar

[6] A. Braides, $\Gamma$-convergence for Beginners, Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar
[7]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-convergence, Springer, Cham, 2014. doi: 10.1007/978-3-319-01982-6.  Google Scholar

[8]

A. BraidesM. ColomboM. Gobbino and M. Solci, Minimizing movements along a sequence of functionals and curves of maximal slope, C. R. Math. Acad. Sci. Paris, 354 (2005), 685-689.  doi: 10.1016/j.crma.2016.04.011.  Google Scholar

[9]

A. BraidesM. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces, Arch. Ration. Mech. Anal., 195 (2010), 469-498.  doi: 10.1007/s00205-009-0215-z.  Google Scholar

[10]

E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for Partial Differential Equations and Applications (C. Baiocchi and J. L. Lions, eds.) Masson, Paris, 29 (1993), 81–98.  Google Scholar

[11]

P. Dondl, T. Frenzel and A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence,, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 68, 45 pp. doi: 10.1051/cocv/2018058.  Google Scholar

[12]

F. Fleissner, $\Gamma$-convergence and relaxation of gradient flows in metric spaces: A minimizing movement approach,, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 28, 29pp. doi: 10.1051/cocv/2017035.  Google Scholar

[13]

F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by de Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci., in press, (2017), 1–30. Google Scholar

[14]

E. Sandier and S. Serfaty, $\Gamma$-convergence of gradient flows with applications to Ginzburg-Landau, Commun. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.  Google Scholar

[15]

A. Tribuzio, Perturbations of minimizing movements and curves of maximal slope, Netw. Heterog. Media, 13 (2018), 423-448.  doi: 10.3934/nhm.2018019.  Google Scholar

show all references

References:
[1]

R. AlicandroA. Braides and M. Cicalese, Phase and anti-phase boundaries in binary discrete systems: A variational viewpoint, Netw. Heterog. Media, 1 (2006), 85-107.  doi: 10.3934/nhm.2006.1.85.  Google Scholar

[2]

F. Almgren and J. E. Taylor, Flat flow is motion by mean curvature for curves with crystalline energy, J. Differential Geom., 42 (1995), 1-22.  doi: 10.4310/jdg/1214457030.  Google Scholar

[3]

F. AlmgrenJ. L. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020.  Google Scholar

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Birkhäuser, Basel, 2008.  Google Scholar

[5]

N. AnsiniA. Braides and J. Zimmer, Minimizing movements for oscillating energies: The critical regime, Proc. Royal Soc. Edin. A, 149 (2019), 719-737.  doi: 10.1017/prm.2018.46.  Google Scholar

[6] A. Braides, $\Gamma$-convergence for Beginners, Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar
[7]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-convergence, Springer, Cham, 2014. doi: 10.1007/978-3-319-01982-6.  Google Scholar

[8]

A. BraidesM. ColomboM. Gobbino and M. Solci, Minimizing movements along a sequence of functionals and curves of maximal slope, C. R. Math. Acad. Sci. Paris, 354 (2005), 685-689.  doi: 10.1016/j.crma.2016.04.011.  Google Scholar

[9]

A. BraidesM. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces, Arch. Ration. Mech. Anal., 195 (2010), 469-498.  doi: 10.1007/s00205-009-0215-z.  Google Scholar

[10]

E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for Partial Differential Equations and Applications (C. Baiocchi and J. L. Lions, eds.) Masson, Paris, 29 (1993), 81–98.  Google Scholar

[11]

P. Dondl, T. Frenzel and A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence,, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 68, 45 pp. doi: 10.1051/cocv/2018058.  Google Scholar

[12]

F. Fleissner, $\Gamma$-convergence and relaxation of gradient flows in metric spaces: A minimizing movement approach,, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 28, 29pp. doi: 10.1051/cocv/2017035.  Google Scholar

[13]

F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by de Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci., in press, (2017), 1–30. Google Scholar

[14]

E. Sandier and S. Serfaty, $\Gamma$-convergence of gradient flows with applications to Ginzburg-Landau, Commun. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.  Google Scholar

[15]

A. Tribuzio, Perturbations of minimizing movements and curves of maximal slope, Netw. Heterog. Media, 13 (2018), 423-448.  doi: 10.3934/nhm.2018019.  Google Scholar

Figure 1.  The dark line represents the graph of $\gamma\mapsto1/a^\gamma$, the light line is the constant $1/a^*$. On the left $\alpha>\beta/2$ so the sup is reached in $\gamma_1^\beta$, on the right $\alpha < \beta/2$ and the sup is reached in $\gamma_1^\alpha$
[1]

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

[2]

Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657

[3]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020328

[4]

Mitsunori Nara, Masaharu Taniguchi. Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 137-156. doi: 10.3934/dcds.2006.16.137

[5]

Wei Wang, Na Sun, Michael K. Ng. A variational gamma correction model for image contrast enhancement. Inverse Problems & Imaging, 2019, 13 (3) : 461-478. doi: 10.3934/ipi.2019023

[6]

Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017

[7]

Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679

[8]

Gilles A. Francfort, Alessandro Giacomini, Alessandro Musesti. On the Fleck and Willis homogenization procedure in strain gradient plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 43-62. doi: 10.3934/dcdss.2013.6.43

[9]

Risei Kano, Yusuke Murase. Solvability of nonlinear evolution equations generated by subdifferentials and perturbations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 75-93. doi: 10.3934/dcdss.2014.7.75

[10]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095

[11]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363

[12]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189

[13]

Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347

[14]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753

[15]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787

[16]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503

[17]

Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks & Heterogeneous Media, 2015, 10 (2) : 233-253. doi: 10.3934/nhm.2015.10.233

[18]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033

[19]

José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025

[20]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (20)
  • HTML views (142)
  • Cited by (0)

Other articles
by authors

[Back to Top]