0000000000341369
AUTHOR
Nathaël Alibaud
showing 2 related works from this author
(Bounded) Traveling combustion fronts with degenerate kinetics
2022
Abstract We consider the propagation of a flame front in a solid periodic medium. It is governed by an equation of Hamilton–Jacobi type, whose front’s velocity depends on the temperature via a nonlinear degenerate kinetic rate. The temperature solves a free boundary problem subject to boundary conditions depending on the front’s velocity itself. We show the existence of nonplanar traveling wave solutions which are bounded and global. Previous results by the same authors (cf. Alibaud and Namah, 2017) were obtained for essentially positively lower bounded kinetics or eventually which have some very weak degeneracy. Here we consider very general degenerate kinetics, including for the first tim…
The Liouville theorem and linear operators satisfying the maximum principle
2020
A result by Courr\`ege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ where $$ \mathcal{L}^{\sigma,b}[u](x)=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} \mu(z). $$ This class of operators coincides with the infinitesimal generators of L\'evy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ i…