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June  2014, 9(2): 217-238. doi: 10.3934/nhm.2014.9.217

Variational evolution of one-dimensional Lennard-Jones systems

Andrea Braides 1, , Anneliese Defranceschi 2, and  Enrico Vitali 3,

1. 

Dipartimento di Matematica, Università di Roma 'Tor Vergata', via della Ricerca Scientifica, 00133 Roma
2. 

Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123 Povo, Italy
3. 

Dipartimento di Matematica 'F. Casorati', Università di Pavia, via Ferrata, 1-27100 Pavia

Received  October 2013 Revised  May 2014 Published  July 2014

We analyze Lennard-Jones systems from the standpoint of variational principles beyond the static framework. In a one-dimensional setting such systems have already been shown to be equivalent to energies of Fracture Mechanics. Here we show that this equivalence can also be given in dynamical terms using the notion of minimizing movements.
Keywords: minimizing movements, Fracture Mechanics., Lennard-Jones systems, variational evolution.
Mathematics Subject Classification: Primary: 35K90, 74A45; Secondary: 49J45, 74R1.
Citation: Andrea Braides, Anneliese Defranceschi, Enrico Vitali. Variational evolution of one-dimensional Lennard-Jones systems. Networks & Heterogeneous Media, 2014, 9 (2) : 217-238. doi: 10.3934/nhm.2014.9.217
References:
[1]

L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191.   Google Scholar

[2]

L. Ambrosio and A. Braides, Energies in $SBV$ and variational models in fracture mechanics,, in Homogenization and Applications to Material Sciences (Nice, (1995), 1.   Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford University Press, (2000).   Google Scholar

[4]

L. Ambrosio and N. Gigli, A user's guide to optimal transport,, in Modelling and Optimisation of Flows on Networks (eds. B. Piccoli and M. Rascle), (2062), 1.   Google Scholar

[5]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics ETH, (2008).   Google Scholar

[6]

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

[7]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence, Lecture Notes in Mathematics, 2094,, Springer, (2014).  doi: 10.1007/978-3-319-01982-6.  Google Scholar

[8]

A. Braides, M. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces,, Arch. Ration. Mech. Anal., 95 (2010), 469.  doi: 10.1007/s00205-009-0215-z.  Google Scholar

[9]

A. Braides, A. J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes,, Arch. Ration. Mech. Anal., 180 (2006), 151.  doi: 10.1007/s00205-005-0399-9.  Google Scholar

[10]

A. Braides and L. Truskinovsky, Asymptotic expansions by Gamma-convergence,, Cont. Mech. Therm., 20 (2008), 21.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[11]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser, (1993).  doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[12]

E. De Giorgi, New problems on minimizing movements,, in Boundary Value Problems for Partial Differential Equations and Applications, (1993), 81.   Google Scholar

[13]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calc. Var. Partial Differential Equations, 39 (2010), 101.  doi: 10.1007/s00526-009-0303-9.  Google Scholar

[14]

M. Gobbino, Gradient flow for the one-dimensional Mumford-Shah functional,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 145.   Google Scholar

[15]

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

show all references

References:
[1]

L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191.   Google Scholar

[2]

L. Ambrosio and A. Braides, Energies in $SBV$ and variational models in fracture mechanics,, in Homogenization and Applications to Material Sciences (Nice, (1995), 1.   Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford University Press, (2000).   Google Scholar

[4]

L. Ambrosio and N. Gigli, A user's guide to optimal transport,, in Modelling and Optimisation of Flows on Networks (eds. B. Piccoli and M. Rascle), (2062), 1.   Google Scholar

[5]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics ETH, (2008).   Google Scholar

[6]

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

[7]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence, Lecture Notes in Mathematics, 2094,, Springer, (2014).  doi: 10.1007/978-3-319-01982-6.  Google Scholar

[8]

A. Braides, M. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces,, Arch. Ration. Mech. Anal., 95 (2010), 469.  doi: 10.1007/s00205-009-0215-z.  Google Scholar

[9]

A. Braides, A. J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes,, Arch. Ration. Mech. Anal., 180 (2006), 151.  doi: 10.1007/s00205-005-0399-9.  Google Scholar

[10]

A. Braides and L. Truskinovsky, Asymptotic expansions by Gamma-convergence,, Cont. Mech. Therm., 20 (2008), 21.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[11]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser, (1993).  doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[12]

E. De Giorgi, New problems on minimizing movements,, in Boundary Value Problems for Partial Differential Equations and Applications, (1993), 81.   Google Scholar

[13]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calc. Var. Partial Differential Equations, 39 (2010), 101.  doi: 10.1007/s00526-009-0303-9.  Google Scholar

[14]

M. Gobbino, Gradient flow for the one-dimensional Mumford-Shah functional,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 145.   Google Scholar

[15]

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

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