Browsing by Author "Guidotti, Patrick"
Now showing 1 - 2 of 2
Results Per Page
Sort Options
Item BOUNDARY DYNAMICS OF A TWO-DIMENSIONAL DIFFUSIVE FREE BOUNDARY PROBLEM(2010-02) Webster, Micah; Guidotti, Patrick; Mathematics; Center for Data, Mathematical, and Computational SciencesNumerous models of industrial processes such as diffusion in glassypolymers or solidification phenomena, lead to general one-phase free boundary value problems with phase onset. In this paper we develop a framework viable to prove global existence and stability of planar solutions to one such multi-dimensional model whose application is in controlled-release pharmaceuticals. We utilize a boundary integral reformulation to allow for the use of maximal regularity. To this effect, we view the operators as pseudo-differential and ex-ploit knowledge of the relevant symbols. Within this framework, we give a local existence and continuous dependence result necessary to prove planar solutions are locally exponentially stable with respect to two-dimensional perturbations.Item Nonlinear stability analysis of a two-dimensional diffusive free boundary problem(2010-01-25) Webster, Micah; Guidotti, Patrick; Mathematics and Computer ScienceWe explore global existence and stability of planar solutions to a multi-dimensional Case II polymer diffusion model which takes the form of a one-phase free boundary problem with phase onset. Due to a particular boundary condition, convergence cannot be expected on the whole domain. A boundary integral formulation derived in [13] is shown to remain valid in the present context and allows us to circumvent this difficulty by restricting the analysis to the free boundary. The integral operators arising in the boundary integral formulation are analyzed by methods of pseudodifferential calculus. This is possible as explicit symbols are available for the relevant kernels. Spectral analysis of the linearization can then be combined with a known principle of linearized stability [12] to obtain local exponential stability of planar solutions with respect to two-dimensional perturbations.