A review on some contributions to perturbation theory, singular limits and well-posedness

Well-posedness of a singular balance law

We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation ∂tu(t, x) + ∂x u 2 (t, x) = −λu(t, x) δ0(x), which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at x = 0. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-...

On a Singular Perturbation Problem Involving a “circular-well” Potential

We study the asymptotic behavior, as a small parameter ε goes to 0, of the minimizers for a variational problem which involves a “circularwell” potential, i.e., a potential vanishing on a closed smooth curve in R2. We thus generalize previous results obtained for the special case of the GinzburgLandau potential.

On some anisotropic singular perturbation problems

We investigate the asymptotic behavior of some anisotropic diffusion problems and give some estimates on the rate of convergence of the solution toward its limit. We also relate this type of elliptic problems to problems set in cylinder becoming unbounded in some directions and show how some information on one type leads to information for the other type and conversely. 1. A model problem The g...

Singular perturbation theory

When we apply the steady-state approximation (SSA) in chemical kinetics, we typically argue that some of the intermediates are highly reactive, so that they are removed as fast as they are made. We then set the corresponding rates of change to zero. What we are saying is not that these rates are identically zero, of course, but that they are much smaller than the other rates of reaction. The st...

Geometric Singular Perturbation Theory