In this paper we prove the existence of a solution for a system of nonlinear parabolic partial differential equations arising from thermoelectric modelling of metallurgical electrodes undergoing a phase change. The model consists of an electromagnetic problem for eddy current computation coupled with a Stefan problem for temperature. The proof uses a regularized problem obtained by truncating the source term in temperature equation. Passing to the limit requires fine a priori estimates leading to compactness.

This paper proposes the use of a variational framework to model fluid wetting dynamics. The central problem of infinite energy dissipation for a moving contact line is dealt with explicitly rather than by introducing a specific microscopic mechanism which removes it. We analyze this modelling approach in the context of the quasi-steady limit, where contact line motion is slower than bulk relaxation. We find that global effects enter into Tanner-type laws which relate line velocity to apparent contact angle through the role that energy dissipation plays in the bulk of the fluid. A comparison is made to the dynamics of lubrication equations that include attractive and repulsive intermolecular interactions. A Galerkin-type approximation method is introduced which leads to reduced-dimensional dynamical descriptions. Computations are conducted using these low-dimensional approximations, and a substantial connection to lubrication equation dynamics is found.

We study the existence of planar flames, in the case of a single-step chemical reaction with volumetric heat losses, with a general reaction term. We prove that for all positive Lewis numbers, and for small values of the heat loss rate parameter, two distinct solutions exist. We also give upper bounds for the flame speed and for the heat loss rate parameter. Moreover, we explicitly compute a lower bound for the unburned gases after reaction.

The parabolic equation \[u_t + u_{xxxx} + u_{xx} = - (|u_x|^\alpha)_{xx}, \qquad \alpha>1\], is studied under the boundary conditions $u_x|_{\partial\Omega}=u_{xxx}|_{\partial\Omega}=0$ in a bounded real interval $\Omega$. Solutions from two different regularity classes are considered: It is shown that unique mild solutions exist locally in time for any $\alpha>1$ and initial data $u_0 \in W^{1,q}(\Omega)$ ($q>\alpha$), and that they are global if $\alpha \le \frac{5}{3}$. Furthermore, from a semidiscrete approximation scheme global weak solutions are constructed for $\alpha < \frac{10}{3}$, and for suitable transforms of such solutions the existence of a bounded absorbing set in $L^1(\Omega)$ is proved for $\alpha \in [2,\frac{10}{3})$. The article closes with some numerical examples which do not only document the roughening and coarsening phenomena expected for thin film growth, but also illustrate our results about absorbing sets.

For superconductors of type II the phenomenon of vortex pinning plays an important role in technological applications. Several models have been proposed for this effect (Kim et al., 1963; Bean, 1964; Bossavit, 1994). In Du et al. (1999) and Prigozhin (1996), some of these models are analyzed. In this work we want to contribute to the analysis for the two-dimensional, rate-independent model proposed in Chapman (2000), which has the special feature that vortex movement and creation is an activated process occurring only when a threshold value of the magnetic field is reached. For analytical studies of related rate-dependent models we refer to Chapman et al. (1996), Schätzle & Styles (1999) and Elliott & Styles (2000).