Compactness, kinetic formulation, and entropies for a problem related to micromagnetics

We carry on the study of (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294-338.) 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 (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294-338.) that the energy-functionals Γ-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.