Augusto Visintin

Models of Phase Transitions

Part I Some nonlinear PDEs: models and PDEs a class of quasilinear parabolic PDEs doubly-nonlinear parabolic PDEs. Part 2 Phase transitions: the Stefan problem generalizations of the Stefan problem.

On Landau-Lifshitz’ equations for ferromagnetism

AbstractAccording to the classical theory of Weiβ, on a microscopic scale a ferromagnetic body is magnetically saturated (|M| =M0: constant) and is composed by uniformly magnetized regions separated by thin transition layers.At equilibrium this corresponds to the minimization of the magnetic energy functional under the above constraint; this problem has at least one solutionM of classH1, which in general is not unique. The evolution is governed by Landau-Lifshitz’ equations

On nonstationary flow through porous media

\begin{gathered} \frac{{\partial M}}{{\partial t}} = \lambda _1 M \times H^e = \lambda _2 M \times \left( {M \times H^e } \right) \left( {\lambda _1 ,\lambda _2 : constant; \lambda _2 > 0} \right) \hfill \\ H^e = - \frac{{\partial e_{mag} \left( M \right)}}{{\partial M}} \left( {e_{mag} : density of magnetic energy} \right) \hfill \\ \end{gathered}

Asymptotic behavior of the Landau--Lifshitz model of ferromagnetism

; these are coupled with Maxwell’s equations. An existence result based on the energy estimate and some complementary properties are proved for the corresponding variational formulation. Finally the magnetostrictive effect is included.

Mathematical Models of Hysteresis

Let Ωbe endowed with a σ—, complete measure and let weakly in . If u(x) is an external point of the closed convex hull of a.e. in Ω, then strongly in cannot oscillate around u(x). Other strong convergence results are proved. Applications to the solution of nonlinear partial differential equaitons and of minimization problems are given.

Evolution problems with hysteresis in the source term

SummarySaturated-unsaturated flow of an incompressible fluid through a porous medium is considered in the case of time-dependent water levels. This corresponds to coupling the mass conservation law with a continuous constitutive condition between water content and pressure. An existence result for the corresponding weak formulation is proved. Finally we study the limit as the constitutive relation degenerates into a maximal monotone graph.

Generalized coarea formula and fractal sets

For materials capable of attaining two phases, the free energy potential is a non-convex functional of the state variables. Absolute and relative minimizers of this functional are related to stable and metastable states, respectively. Metastability explains the undercooling required for solid nucleation.

A model for hysteresis of distributed systems

According to the classical theory of Weiss, Landau, and Lifshitz, on a microscopic scale a ferromagnetic body is magnetically saturated (i.e., |M| =ℳ: constant) and consists of regions in which the magnetization is uniform, separated by thin transition layers. Any stationary configuration corresponds to a minimum point of an energy functional in which a small parameterε is present. The asymptotic behavior asε → 0 is studied here. It is easy to see that any sequence of minimizers contains a subsequenceMεj that converges to a fieldM. By means of a Γ-limit procedure it is shown that this fieldM is a minimizer of a new functional containing a term proportional to the area of the surfaces separating different domains of uniform magnetization. TheC1,γ-regularity of these surfaces, forγ < 1/2, is also proved under suitable assumptions for the external magnetic field.