Singularities of divergence-free vector fields with values into S1 or S2: Applications to micromagnetics
Confluentes Mathematici, Volume 4 (2012) no. 3.

In this survey, we present several results on the regularizing effect, rigidity and approximation of 2D unit-length divergence-free vector fields. We develop the concept of entropy (coming from scalar conservation laws) in order to analyze singularities of such vector fields. In particular, based on entropies, we characterize lower semicontinuous line-energies in 2D and we study by Γ-convergence method the associated regularizing models (like the 2D Aviles–Giga and the 3D Bloch wall models). We also present some applications to the analysis of pattern formation in micromagnetics. In particular, we describe domain walls in the thin ferromagnetic films (e.g. symmetric Néel walls, asymmetric Néel walls, asymmetric Bloch walls) together with interior and boundary vortices.

Published online:
DOI: 10.1142/S1793744212300012
Radu Ignat 1

1
@article{CML_2012__4_3_A1_0,
     author = {Radu Ignat},
     title = {Singularities of divergence-free vector fields with values into $S^1$ or $S^2$: {Applications} to micromagnetics},
     journal = {Confluentes Mathematici},
     publisher = {World Scientific Publishing Co Pte Ltd},
     volume = {4},
     number = {3},
     year = {2012},
     doi = {10.1142/S1793744212300012},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744212300012/}
}
TY  - JOUR
AU  - Radu Ignat
TI  - Singularities of divergence-free vector fields with values into $S^1$ or $S^2$: Applications to micromagnetics
JO  - Confluentes Mathematici
PY  - 2012
VL  - 4
IS  - 3
PB  - World Scientific Publishing Co Pte Ltd
UR  - https://cml.centre-mersenne.org/articles/10.1142/S1793744212300012/
DO  - 10.1142/S1793744212300012
LA  - en
ID  - CML_2012__4_3_A1_0
ER  - 
%0 Journal Article
%A Radu Ignat
%T Singularities of divergence-free vector fields with values into $S^1$ or $S^2$: Applications to micromagnetics
%J Confluentes Mathematici
%D 2012
%V 4
%N 3
%I World Scientific Publishing Co Pte Ltd
%U https://cml.centre-mersenne.org/articles/10.1142/S1793744212300012/
%R 10.1142/S1793744212300012
%G en
%F CML_2012__4_3_A1_0
Radu Ignat. Singularities of divergence-free vector fields with values into $S^1$ or $S^2$: Applications to micromagnetics. Confluentes Mathematici, Volume 4 (2012) no. 3. doi : 10.1142/S1793744212300012. https://cml.centre-mersenne.org/articles/10.1142/S1793744212300012/

[1] F. Alouges, T. Rivière and S. Serfaty, Néel and cross-tie wall energies for planar micromagnetic configurations, ESAIM Control Optim. Calc. Var. 8 (2002) 31–68.

[2] L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane, Calc. Var. Partial Differential Equations 9 (1999) 327–255.

[3] L. Ambrosio, B. Kirchheim, M. Lecumberry and T. Rivière, On the rectifiability of defect measures arising in a micromagnetics model, in Nonlinear Problems in Mathe- matical Physics and Related Topics, II, Int. Math. Ser., Vol. 2 (Kluwer/Plenum, 2002), pp. 29–60.

[4] P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations, in Miniconference on Geometry and Partial Differential Equations, 2 (Canberra, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 12 (Austral. Nat. Univ., 1987), pp. 1–16.

[5] P. Aviles and Y. Giga, The distance function and defect energy, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 923–938.

[6] P. Aviles and Y. Giga, On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg–Landau type energy for gradient fields, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 1–17.

[7] F. Béthuel, H. Brezis and F. Hélein, Ginzburg–Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, Vol. 13 (Birkhäuser, 1994).

[8] F. Bethuel and X. M. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80 (1988) 60–75.

[9] S. Bianchini, C. De Lellis and R. Robyr, SBV regularity for Hamilton–Jacobi equations in Rn , Arch. Rational Mech. Anal. 200 (2011) 1003–1021.

[10] J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86.

[11] J. Bourgain, H. Brezis and P. Mironescu, H 1 2 maps with values into the circle: Minimal connections, lifting, and the Ginzburg–Landau equation, Publ. Math. Inst. Hautes Études Sci. 99 (2004) 1–115.

[12] H. Brezis, P. Mironescu and A. C. Ponce, W 1,1 -maps with values into S1 , in Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., Vol. 368 (Amer. Math. Soc., 2005), pp. 69–100.

[13] W. F. Brown, Micromagnetics (Wiley Interscience, 1963).

[14] L. A. Caffarelli and M. G. Crandall, Distance functions and almost global solutions of eikonal equations, Comm. Partial Differential Equations 35 (2010) 391–414.

[15] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial energy, J. Chem. Phys. 28 (1958) 258–267.

[16] G. Carbou, Thin layers in micromagnetism, Math. Models Methods Appl. Sci. 11 (2001) 1529–1546.

[17] S. Conti and C. De Lellis, Sharp upper bounds for a variational problem with singular perturbation, Math. Ann. 338 (2007) 119–146.

[18] J. D´avila and R. Ignat, Lifting of BV functions with values in S1 , C. R. Math. Acad. Sci. Paris 337 (2003) 159–164.

[19] C. De Lellis, An example in the gradient theory of phase transitions, ESAIM Control Optim. Calc. Var. 7 (2002) 285–289.

[20] C. De Lellis and F. Otto, Structure of entropy solutions to the eikonal equation, J. Eur. Math. Soc. 5 (2003) 107–145.

[21] A. DeSimone, H. Knüpfer and F. Otto, 2d stability of the Néel wall, Calc. Var. Partial Differential Equations 27 (2006) 233–253.

[22] A. DeSimone, R. V. Kohn, S. Müller and F. Otto, Magnetic microstructures — a paradigm of multiscale problems, in ICIAM 99 (Edinburgh), (Oxford Univ. Press, 2000), pp. 175–190.

[23] A. Desimone, R. V. Kohn, S. Müller and F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1408–1460.

[24] A. Desimone, R. V. Kohn, S. Müller and F. Otto, Repulsive interaction of Néel walls, and the internal length scale of the cross-tie wall, Multiscale Model. Simul. 1 (2003) 57–104.

[25] A. DeSimone, R. V. Kohn, S. Müller, F. Otto and R. Schäfer, Two-dimensional mod- elling of soft ferromagnetic films, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001) 2983–2991.

[26] A. DeSimone, S. Müller, R. V. Kohn and F. Otto, A compactness result in the gradient theory of phase transitions, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 833–844.

[27] A. DeSimone, S. Müller, R. V. Kohn and F. Otto, Recent analytical developments in micromagnetics, in The Science of Hysteresis, Vol. 2 (Elsevier, 2005), pp. 269–381.

[28] L. Döring, R. Ignat and F. Otto, Asymmetric domain walls of small angle in micro- magnetics, preprint.

[29] L. Döring, R. Ignat and F. Otto, Cross-over from symmetric to asymmetric transition layers in micromagnetics, preprint.

[30] M. Giaquinta, G. Modica and J. Souˇcek, Cartesian Currents in the Calculus of Vari- ations, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 38 (Springer-Verlag, 1998).

[31] Y. Giga, M. Kubo and Y. Tonegawa, Magnetic clusters and fold energies, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 23–40.

[32] F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988) 110–125.

[33] A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstruc- tures (Springer-Verlag, 1998).

[34] R. Ignat, On an open problem about how to recognize constant functions, Houston J. Math. 31 (2005) 285–304.

[35] R. Ignat, The space BV(S2 , S1 ): Minimal connection and optimal lifting, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 283–302.

[36] R. Ignat, A Γ-convergence result for Néel walls in micromagnetics, Calc. Var. Partial Differential Equations 36 (2009) 285–316.

[37] R. Ignat, A survey of some new results in ferromagnetic thin films, in Séminaire: Équations aux Dérivées Partielles. 2007–2008, Sémin. Équ. Dériv. Partielles, pages Exp. No. VI, 21 (École Polytech., 2009).

[38] R. Ignat, Gradient vector fields with values into S1 , C. R. Math. Acad. Sci. Paris 349 (2011) 883–887.

[39] R. Ignat, Two-dimensional unit-length vector fields of vanishing divergence, J. Funct. Anal. 262 (2012) 3465–3494.

[40] R. Ignat and H. Knüpfer, Vortex energy and 360◦ Néel walls in thin-film micromag- netics, Comm. Pure Appl. Math. 63 (2010) 1677–1724.

[41] R. Ignat and M. Kurzke, An effective model for boundary vortices in thin-film micro- magnetics, in preparation.

[42] R. Ignat and B. Merlet, Lower bound for the energy of Bloch walls in micromagnetics, Arch. Rational Mech. Anal. 199 (2011) 369–406.

[43] R. Ignat and B. Merlet, Entropy method for line-energies, Calc. Var. Partial Differ- ential Equations 44 (2012) 375–418.

[44] R. Ignat and R. Moser, A zigzag pattern in micromagnetics, J. Math. Pures Appl. 98 (2012) 139–159.

[45] R. Ignat and F. Otto, A compactness result in thin-film micromagnetics and the optimality of the Néel wall, J. Eur. Math. Soc. 10 (2008) 909–956.

[46] R. Ignat and F. Otto, A compactness result of Landau state in thin-film micromag- netics, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011) 247–282.

[47] P.-E. Jabin, F. Otto and B. Perthame, Line-energy Ginzburg–Landau models: Zero- energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202.

[48] P.-E. Jabin and B. Perthame, Compactness in Ginzburg–Landau energy by kinetic averaging, Comm. Pure Appl. Math. 54 (2001) 1096–1109.

[49] R. L. Jerrard, Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal. 30 (1999) 721–746.

[50] W. Jin and R. V. Kohn, Singular perturbation and the energy of folds, J. Nonlinear Sci. 10 (2000) 355–390.

[51] R. V. Kohn and V. V. Slastikov, Another thin-film limit of micromagnetics, Arch. Rational Mech. Anal. 178 (2005) 227–245.

[52] S. N. Kruˇzkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81 (1970) 228–255.

[53] M. Kurzke, Boundary vortices in thin magnetic films, Calc. Var. Partial Differential Equations 26 (2006) 1–28.

[54] M. Kurzke, A nonlocal singular perturbation problem with periodic well potential, ESAIM Control Optim. Calc. Var. 12 (2006) 52–63.

[55] F. H. Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math. 52 (1999) 737–761.

[56] C. Melcher, The logarithmic tail of Néel walls, Arch. Rational Mech. Anal. 168 (2003) 83–113.

[57] C. Melcher, Logarithmic lower bounds for Néel walls, Calc. Var. Partial Differential Equations 21 (2004) 209–219.

[58] P. Mironescu, Lifting of S1 -valued maps in sums of Sobolev spaces, preprint.

[59] P. Mironescu, S1 -valued Sobolev mappings, Sovrem. Mat. Fundam. Napravl. 35 (2010) 86–100.

[60] R. Moser, Ginzburg–Landau vortices for thin ferromagnetic films, AMRX Appl. Math. Res. Express 1 (2003) 1–32.

[61] R. Moser, Boundary vortices for thin ferromagnetic films, Arch. Rational Mech. Anal. 174 (2004) 267–300.

[62] F. Murat, Compacité par compensation: Condition nécessaire et suffisante de conti- nuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) 69–102.

[63] C. B. Muratov and V. V. Osipov, Theory of 360◦ domain walls in thin ferromagnetic films, J. Appl. Phys. 104 (2008) 053908.

[64] M. Ortiz and G. Gioia, The morphology and folding patterns of buckling-driven thin- film blisters, J. Mech. Phys. Solids 42 (1994) 531–559.

[65] F. Otto, Cross-over in scaling laws: A simple example from micromagnetics, in Proc. of the Int. Congress of Mathematicians, Vol. III (Beijing, 2002) (Higher Ed. Press, 2002), pp. 829–838.

[66] A. Poliakovsky, Upper bounds for singular perturbation problems involving gradient fields, J. Eur. Math. Soc. 9 (2007) 1–43.

[67] T. Rivière, Dense subsets of H 1 2 (S 2 ,S 1 ), Ann. Global Anal. Geom. 18 (2000) 517–528.

[68] T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 (2001) 294–338.

[69] T. Rivière and S. Serfaty, Compactness, kinetic formulation, and entropies for a prob- lem related to micromagnetics, Comm. Partial Differential Equations 28 (2003) 249– 269.

[70] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998) 379–403.

[71] E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg–Landau Model, Progress in Nonlinear Differential Equations and their Applications, Vol. 70 (Birkhäuser, 2007).

[72] L. Tartar. Compensated compactness and applications to partial differential equa- tions, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., Vol. 39 (Pitman, 1979), pp. 136–212.

[73] H. A. M. van den Berg, Self-consistent domain theory in soft-ferromagnetic media. II, Basic domain structures in thin film objects, J. Appl. Phys. 60 (1986) 1104–1113.

Cited by Sources: