Kinetic methods for Line–energy Ginzburg–Landau models

Niveau: Supérieur
Kinetic methods for Line–energy Ginzburg–Landau models Pierre-Emmanuel Jabin and Benoıt Perthame Departement de Mathematiques et Applications, UMR8553, Ecole Normale Superieure, 45, rue d'Ulm, 75230 Paris Cedex 05, France July 17, 2002 Abstract. A class of variational problems arising in thin micromagnetic film or in the gradient theory of phase transitions exhibit an hyperbolic behavior, a surprising property being given their natural elliptic structure. These two–dimensional Ginzburg–Landau problems are, for in- stance, characterized by energy density concentrations on a one–dimensional set - comparable to a steady shock wave. Here we review how methods based on kinetic formulations can help to understand some feautures of this broad and fascinating class of problems. Especially we deduce a general regularity result and also we characterize the zero-energy states and the domains where they can occur. Key words. Ginzburg–Landau energy, vortices, kinetic formulation, averaging lemmas, Sobolev spaces. AMS Class. Numbers. 35B65, 35J60, 35L65, 74G65, 82D30. Contents 1 Typical examples 1 2 kinetic formulation 3 3 A generic regularity result 4 4 Vortices and zero energy states 6 1 Typical examples Among the wide subject of Ginzburg-Landau variational problems, a typical problem is to study the limit as the parameter ? vanishes, for divergence free functions in R2 with a finite Ginzburg- Landau energy.

• thus does


• energy states


• averaging lemmas


• ginzburg–landau problems


• line–energy ginzburg–landau


• vlasov equation


• kinetic formulation


• ginzburg-landau variational


• scalar conservation