M. Cristina Depassier

On the linear stability theory of Bénard–Marangoni convection

The linear stability of a fluid bounded above by a free deformable surface is studied numerically. When the heat flux is fixed on the free surface and the lower surface is plane and isothermic, oscillatory instabilities, which may occur at lower values of the Rayleigh number than the critical value for the onset of steady convection, are found.

Upper and lower bounds for eigenvalues of nonlinear elliptic equations: I. The lowest eigenvalue

We give a method for finding bounds for the lowest eigenvalue of nonlinear elliptic equations with monotone, local, nonlinearities. This is an extension to nonlinear problems of Barta’s method for linear elliptic operators.

Variational Principles for the Speed of Traveling Fronts of Reaction–Diffusion Equations

Abstract The 1D nonlinear diffusion equation has been used to model a variety of phenomena in different fields, e.g. population dynamics, flame propagation, combustion theory, chemical kinetics and many others. After the work of Fisher [Ann. Eugenics 7 (1937) 355] and Kolmogorov et al. [Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique Vol. 1] in the late 1930s, there has been a vast literature on the study of the propagation of localized initial disturbances. The purpose of this review is to present a rather recent variational characterization of the minimal speed of propagation, together with some of its consequences and applications. We consider the 1D reaction–diffusion equation as well as several extensions.

Rigorous results for the minimal speed of Kolmogorov–Petrovskii–Piscounov monotonic fronts with a cutoffa)

We study the effect of a cutoff on the speed of pulled fronts of the one-dimensional reaction diffusion equation. To accomplish this, we first use variational techniques to prove the existence of a heteroclinic orbit in phase space for traveling wave solutions of the corresponding reaction diffusion equation under conditions that include discontinuous reaction profiles. This existence result allows us to prove rigorous upper and lower bounds on the minimal speed of monotonic fronts in terms of the cut-off parameter ɛ. From these bounds we estimate the range of validity of the Brunet–Derrida formula for a general class of reaction terms.

Variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion

We study the asymptotic speed of traveling fronts of the scalar reaction diffusion for positive reaction terms and with a diffusion coefficient depending nonlinearly on the concentration and on its gradient. We restrict our study to diffusion coefficients of the form

A reversed Poincaré inequality for monotone functions

D(u,u_x) = m u^{m-1} u_x^{m(p-2)}

Rigorous results for the speed of Kolmogorov-Petrovskii-Piscounov fronts with a cutoff

D(u,ux)=mum-1uxm(p-2) for which existence and convergence to traveling fronts have been established. We formulate a variational principle for the asymptotic speed of the fronts. Upper and lower bounds for the speed valid for any