arXiv subjects
Publications and source records attributed to Christoph Walker.
The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.
Touchdown is shown to be the only possible finite time singularity that may take place in a free boundary problem modeling a three-dimensional microelectromechanical system. The proof relies on the energy structure of the problem and uses smoothing effects of the semigroup generated in $L^1$ by the bi-Laplacian with clamped boundary conditions.
We address the well-posedness of the Muskat problem in a periodic geometry and in a setting which allows us to consider general initial and boundary data, gravity effects, as well as surface tension effects. In the absence of surface tension we prove that the Rayleigh-Taylor condition identifies a domain of parabolicity for the Muskat problem. This property is used to establish the well-posedness of the problem. In the presence of surface tension effects the Muskat problem is of parabolic type for general initial and boundary data. As a bi-product of our analysis we obtain that Dirichlet-Neumann type operators associated with certain diffraction problems are negative generators of strongly continuous and analytic semigroups in the scale of small Hölder spaces.
We give a proof for the asymptotic exponential stability of equilibria of quasilinear parabolic evolution equations in admissible interpolation spaces.
We review some results on abstract linear and nonlinear population models with age and spatial structure. The results are mainly based on the assumption of maximal $L_p$-regularity of the spatial dispersion term. In particular, this property allows us to characterize completely the generator of the underlying linear semigroup and to give a simple proof of asynchronous exponential growth of the semigroup. Moreover, maximal regularity is also a powerful tool in order to establish the existence of nontrivial positive equilibrium solutions to nonlinear equations by fixed point arguments or bifurcation techniques. We illustrate the results with examples.
A semilinear parabolic equation with constraint modeling the dynamics of a microelectromechanical system (MEMS) is studied. In contrast to the commonly used MEMS model, the well-known pull-in phenomenon occurring above a critical potential threshold is not accompanied by a breakdown of the model, but is recovered by the saturation of the constraint for pulled-in states. It is shown that a maximal stationary solution exists and that saturation only occurs for large potential values. In addition, the existence, uniqueness, and large time behavior of solutions to the evolution equation are studied.
An idealized electrostatically actuated microelectromechanical system (MEMS) involving an elastic plate with a heterogeneous dielectric material is considered. Starting from the electrostatic and mechanical energies, the governing evolution equations for the electrostatic potential and the plate deflection are derived from the corresponding energy balance. This leads to a free boundary transmission problem due to a jump of the dielectric permittivity across the interface separating elastic plate and free space. Reduced models retaining the influence of the heterogeneity of the elastic plate under suitable assumptions are obtained when either the elastic's plate thickness or the aspect ratio of the device vanishes.
In the past fifteen years mathematical models for microelectromechanical systems (MEMS) have been the subject of several studies, in particular due to the interesting qualitative properties they feature. Still most research is devoted to an illustrative, but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes, the so-called vanishing (or small) aspect ratio model. The analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey.
In this paper the model for a highly viscous droplet sliding down an inclined plane is analyzed. It is shown that, provided the slope is not too steep, the corresponding moving boundary problem possesses classical solutions. Well-posedness is lost when the relevant linearization ceases to be parabolic. This occurs above a critical incline which depends on the shape of the initial wetted region as well as on the liquid's mass. It is also shown that translating circular solutions are asymptotically stable if the kinematic boundary condition is given by an affine functionand provided the incline is small.
We consider a nonlinear integro-differential equation for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The equation can be written as a quasilinear Cauchy problem. For bounded reaction rates we prove global existence and uniqueness of classical solutions by means of evolution operator theory. We also prove global existence of weak solutions for unbounded reaction rates by a compactness argument.
We consider a model for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The model consists of an ordinary differential equation for the prion monomers and a hyperbolic nonlinear differential equation with integral terms for the prion polymers and was shown to possess global weak solutions for unbounded reaction rates [11]. Here we prove the uniqueness of weak solutions.
A free boundary problem modeling a microelectromechanical system (MEMS) consisting of a fixed ground plate and a deformable top plate is considered, the plates being held at different electrostatic potentials. It couples a second order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The validity of the model is expected to break down in finite time when the applied voltage exceeds a certain value, a finite time singularity occurring then. This result, already known for non-positive initial configurations of the top plate, is here proved for arbitrary ones and thus now includes, in particular, snap-through instabilities.
So far most studies on mathematical models for microelectromechanical systems (MEMS) are focused on the so-called small aspect ratio model which is a wave or beam equation with a singular source term. It is formally derived by setting the aspect ratio equal to zero in a more complex model involving a moving boundary. A rigorous justification of this derivation is provided here when bending is taken into account.
We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter $λ$ proportional to the square of the applied voltage. We prove that this free boundary problem is locally well-posed in time and that for small values of $λ$ solutions exist globally in time. We also derive the existence of a branch of asymptotically stable stationary solutions for small values of $λ$ and non-existence of stationary solutions for large values thereof, the latter being restricted to a disc-shaped plate.
We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic Hölder spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.
A variational approach is employed to find stationary solutions to a free boundary problem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are constructed using a constrained minimization problem. A by-product is the existence of at least two stationary solutions for some values of the applied voltage.
We consider a free boundary problem modeling electrostatic microelectromechanical systems. The model consists of a fourth-order damped wave equation for the elastic plate displacement which is coupled to an elliptic equation for the electrostatic potential. We first review some recent results on existence and non-existence of steady-states as well as on local and global well-posedness of the dynamical problem, the main focus being on the possible touchdown behavior of the elastic plate. We then investigate the behavior of the solutions in the time singular limit when the ratio between inertial and damping effects tends to zero.
Lubrication equations for a surfactant-driven flow of a thin layer of fluid are considered with a singular surfactant-dependent surface tension. It is shown that there exists a bounded traveling wave solution which connects a fully surfactant-coated film to an uncoated film. The regularity of the traveling wave depends on whether or not surface diffusion of the insoluble surfactant is included.