We study a variational formulation for a Stefan problem in two adjoining bodies, when the heat conductivity of one of them becomes infinitely large. We study the «concentrated capacity» model arising in the limit, and we justify it by an asymptotic analysis, which is developed in the general framework of the abstract evolution equations of monotone type.

The oriented movement of biological cells or organisms in response to a chemical gradient is called chemotaxis. The most interesting situation related to self-organization phenomenon takes place when the cells detect and response to a chemical which is secreted by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many particularized models have been proposed to describe the aggregation phase of this process. Most of...

We study vortices for solutions of the perturbed Ginzburg–Landau equations $\Delta u+(u/{\epsilon }^2){\left(1-\right|u|}^2)=f_{\epsilon }$ where $f_{\epsilon }$ is estimated in $L^2$. We prove upper bounds for the Ginzburg–Landau energy in terms of $\parallel f_{\epsilon }{\parallel }_{L^2}$, and obtain lower bounds for $\parallel f_{\epsilon }{\parallel }_{L^2}$ in terms of the vortices when these form “unbalanced clusters” where ${\sum }_i{d}_i^2\ne {\left({\sum }_i{d}_i\right)}^2$. These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow one to study various phenomena occurring in this flow, including...

We deduce from the first part of this paper [S1] estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow us to study various phenomena occurring in this flow, including vortex collisions; they allow in particular extending the dynamical law of vortices past collision times.

We discuss the asymptotics of the parabolic Ginzburg-Landau equation in dimension $N\ge 2.$ Our only asumption on the initial datum is a natural energy bound. Compared to the case of “well-prepared” initial datum, this induces possible new energy modes which we analyze, and in particular their mutual interaction. The two dimensional case is qualitatively different and requires a separate treatment.