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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.
@article{CML_2012__4_3_A1_0,
author = {Ignat, Radu},
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 = {http://geodesic.mathdoc.fr/articles/10.1142/S1793744212300012/}
}
TY - JOUR AU - Ignat, Radu 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 - http://geodesic.mathdoc.fr/articles/10.1142/S1793744212300012/ DO - 10.1142/S1793744212300012 LA - en ID - CML_2012__4_3_A1_0 ER -
%0 Journal Article %A Ignat, Radu %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 http://geodesic.mathdoc.fr/articles/10.1142/S1793744212300012/ %R 10.1142/S1793744212300012 %G en %F CML_2012__4_3_A1_0
Ignat, Radu. Singularities of divergence-free vector fields with values into $S^1$ or $S^2$: Applications to micromagnetics. Confluentes Mathematici, Tome 4 (2012) no. 3. doi: 10.1142/S1793744212300012
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