Effect of film morphology on the magnetic properties for Nd-Fe-B thin films

The microstructure and magnetic properties of Nd-Fe-B thin films with a particulate structure were investigated. The nominal thickness of the Nd-Fe-B layer (t_N) was varied from 2 to 50 nm on a (0 0 1) Mo buffer layer. The films were grown with their c-axis perpendicular to the plane, and the morphology of the film with t_N=2 nm was shown to be particulate from an atomic force microscope image. The slope of the initial magnetization curve became steeper by increasing the t_N value in the initial magnetization curve, indicating that the film morphology composed of single domain particles changed to that of multi-domain particles with growth. The film with t_N=8 nm, which had a structure consisting of a mixture of single and multiple domain particles, showed the maximum value of the coercivity measured in the direction perpendicular to the film plane (H_c) as 19.5 kOe.     

Construction of weak solutions of a certain stochastic Navier–Stokes equation

We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.     

Hardy space of exact forms on

We show that the Hardy space of divergence-free vector fields on has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of . Using the duality result we prove a ``div-curl" type theorem: for in , is equivalent to a -type norm of , where the supremum is taken over all with This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on , study their atomic decompositions and dual spaces, and establish ``div-curl" type theorems on .