On the singular limit in a phase field model of phase transitions
On the singular limit of solutions to the Cox-Ingersoll-Ross interest rate model with stochastic volatility
In this paper we are interested in term structure models for pricing zero coupon bonds under rapidly oscillating stochastic volatility. We analyze solutions to the generalized Cox–Ingersoll–Ross two factors model describing clustering of interest rate volatilities. The main goal is to derive an asymptotic expansion of the bond price with respect to a singular parameter representing the fast scale for the stochastic volatility process. We derive the second order asymptotic expansion of a solution...
On the Stefan problem with a small parameter
We consider the multidimensional two-phase Stefan problem with a small parameter κ in the Stefan condition, due to which the problem becomes singularly perturbed. We prove unique solvability and a coercive uniform (with respect to κ) estimate of the solution of the Stefan problem for t ≤ T₀, T₀ independent of κ, and the existence and estimate of the solution of the Florin problem (Stefan problem with κ = 0) in Hölder spaces.
On the structure of layers for singularly perturbed equations in the case of unbounded energy
We consider singular perturbation variational problems depending on a small parameter . The right hand side is such that the energy does not remain bounded as . The asymptotic behavior involves internal layers where most of the energy concentrates. Three examples are addressed, with limits elliptic, parabolic and hyperbolic respectively, whereas the problems with are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after integrating...
On the structure of layers for singularly perturbed equations in the case of unbounded energy
We consider singular perturbation variational problems depending on a small parameter ε. The right hand side is such that the energy does not remain bounded as ε → 0. The asymptotic behavior involves internal layers where most of the energy concentrates. Three examples are addressed, with limits elliptic, parabolic and hyperbolic respectively, whereas the problems with ε > 0 are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after...
On the structure of solutions to elliptic problems with strong anisotropy.
Optimal interface error estimates for the mean curvature flow
P-adaptive Hermite methods for initial value problems∗
We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...
P-adaptive Hermite methods for initial value problems
We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...
P-adaptive Hermite methods for initial value problems∗
We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...
Partial differential equations with piecewise constant delay.
Peak solutions for an elliptic system of FitzHugh-Nagumo type
The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.
Permanence under strong aggressions is possible
Perturbation antisymétrique et oscillations dans des équations paraboliques
L'objet de cet exposé est l'étude d'équations d'évolution de type parabolique, périodiques, que l'on pénalise par un terme linéaire, antisymétrique. Par application des méthodes de S. Schochet pour le cas hyperbolique, on obtient un développement asymptotique des solutions de telles équations. La méthode suivie consiste à étudier l'influence de fortes oscillations en temps dans des systèmes paraboliques. Cette théorie est appliquée à deux systèmes décrivant le comportement de fluides géophysiques,...
Perturbation of a biharmonic eigenvalue problem
Perturbations singulières coercives : réduction à des perturbations régulières et applications
Perturbations singulières de problèmes aux limites, non linéaires, «paraboliques dégénérés-hyperboliques»
Perturbations singulières d'équations hyperboliques du second ordre non linéaires
Perturbations singulières des problèmes de point selle et préconditionnement du problème de Stokes
Perturbations singulières et noyaux de Poisson