Compactness, kinetic formulation, and entropies for a problem related to micromagnetics. (English) Zbl 1094.35125

Rivière, Tristan; Serfaty, Sylvia

Commun. Partial Differ. Equations 28, No. 1-2, 249-269 (2003).

Summary: We carry on the study of a former paper in [Commun. Pure Appl. Math. 54, No. 3, 294–338 (2001; Zbl 1031.35142)] on the asymptotics of a family of energy-functionals related to micromagnetics. We prove compactness for families of uniformly bounded energies releasing the LBP condition we had previously set. Such families converge to unit-valued divergence-free vector-fields that are tangent to the boundary of the domain, and we found in the paper cited above that the energy-functionals \(\Gamma\)-converge to a limiting jump-energy of such configurations. We examine the behavior of certain truncated fields which serve to construct “entropies,” and to provide an improved lower bound. We give a kinetic formulation of the problem, and show that the limiting divergence-free problem is supplemented, in the case of minimizers, with a sign condition which can in turn, using the kinetic formulation, be interpreted as an entropy condition that plays a role in uniqueness questions.

MSC:

35Q60	PDEs in connection with optics and electromagnetic theory
35A15	Variational methods applied to PDEs
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
35J35	Variational methods for higher-order elliptic equations
82D40	Statistical mechanics of magnetic materials

Citations:

Zbl 1031.35142

References:

[1]	Alouges F, ESAIM Control Opt and Calc Var 8 pp 31– (2002) · Zbl 1092.82047
[2]	Ambrosio L, Calc Var PDE 9 (4) pp 327– (1999) · Zbl 0960.49013
[3]	Bethuel F, J Func Anal 80 (1) pp 60– (1988) · Zbl 0657.46027
[4]	DeSimone A, Proc Royal Soc Edinburgh 131 (4) pp 833– (2001) · Zbl 0986.49009
[5]	DiPerna R, Analyse non linéaire 8 pp 271– (1991)
[6]	Jin W, Journal of Nonlinear Science. 10 (3) pp 355– (2000) · Zbl 0973.49009
[7]	Jabin PE, Ann Scuola Normale Sup Pisa Cl Sci 5 pp 187– (2002)
[8]	Jabin PE, Comm Pure Appl Math 54 (9) pp 1096– (2001) · Zbl 1124.35312
[9]	Lions PL, J Amer Math Soc 7 pp 169– (1994)
[10]	Lecumberry M, Calc of Var PDE 15 (3) pp 389– (2002) · Zbl 1021.35023
[11]	Murat F., J Math Pures Appl 60 pp 309– (1981)
[12]	Perthame B., J Math Pures Appl 77 pp 1055– (1998) · Zbl 0919.35088
[13]	Rivière T, Comm Pure Appl Math 54 (3) pp 294– (2001) · Zbl 1031.35142