Compactness in Ginzburg-Landau energy by kinetic averaging.(English)Zbl 1124.35312

Summary: We consider a Ginzburg-Landau energy for two-dimensional, divergence-free fields, which appear in the gradient theory of phase transition, for instance. We prove that as the relaxation parameter vanishes, families of such fields with finite energy are compact in $$L^p(\Omega)$$. Our proof is based on a kinetic interpretation of the entropies that were introduced by DeSimone, Kohn, Müller, and Otto. The so-called kinetic averaging lemmas allow us to generalize their compactness results. Also, the method yields a kinetic equation for the limit where the right-hand side is an unknown kinetic defect bounded measure from which we deduce some Sobolev regularity. This measure also satisfies some cancellation properties depending on its local regularity, which seem to indicate several levels of singularities in the limit.

MSC:

 35J60 Nonlinear elliptic equations 35B25 Singular perturbations in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text:

References:

 [1] Ambrosio, Calc Var Partial Differential Equations 9 pp 327– (1999) · Zbl 0960.49013 [2] Aviles, Proc Roy Soc Edinburgh Sect A 129 pp 1– (1999) · Zbl 0923.49008 [3] ; ; Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and Their Applications, 13. Birkhäuser, Boston, 1994. [4] ; ; Kinetic equations and asymptotic theory. Series in Applied Mathematics, 4. Gauthiers-Villars, Paris; North-Holland, Amsterdam, 2000. [5] DeSimone, Proc. Roy. Soc. Edinburgh [6] DeSimone, ICIAM [7] DiPerna, Ann Inst H Poincaré Anal Non Linéaire 8 pp 271– (1991) · Zbl 0763.35014 [8] Jin, J Nonlinear Sci 10 pp 355– (2000) · Zbl 0973.49009 [9] Lions, J Amer Math Soc 7 pp 169– (1994) [10] Lions, Comm Math Phys 163 pp 415– (1994) · Zbl 0799.35151 [11] Perthame, J Math Pures Appl (9) 77 pp 1055– (1998) · Zbl 0919.35088 [12] Perthame, Ann Sci École Norm Sup (4) 31 pp 591– (1998) · Zbl 0956.45010 [13] ; Limiting domain wall energy in micromagnetism. Preprint, 2000.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.