Instabilities in a combustion model with two free interfaces

Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation

We consider a free interface problem which stems from a solid-gas model in combustion with pattern formation. We derive a third-order, fully nonlinear, self-consistent equation for the flame front. Asymptotic methods reveal that the interface approaches a solution of the Kuramoto-Sivashinsky equation. Numerical results are presented which illustrate the dynamics.

Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface

In this paper we analyze the stability of the traveling wave solution for an ignition-temperature, first-order reaction model of thermo-diffusive combustion, in the case of high Lewis numbers (${\rm Le} >1$). The system of two parabolic PDEs is characterized by a free interface at which ignition temperature $Θ_i$ is reached. We turn the model to a fully nonlinear problem in a fixed domain. When the Lewis number is large, we define a bifurcation parameter $m=Θ_i/(1-Θ_i)$ and a perturbation parameter $\varepsilon= 1/{\rm Le}$. The main result is the existence of a critical value $m^c(\varepsilon…

(Bounded) Traveling Combustion Fronts With Degenerate Kinetics

We consider the propagation of a flame front in a solid periodic medium. The model is governed by a free boundary system in which the front's velocity depends on the temperature via a kinetic rate which may degenerate. We show the existence of travelling wave solutions which are bounded and global. Previous results by the same authors (cf. Alibaud, Namah, On the propagation of a periodic flame front by an Arrhenius kinetic,{\it Interfaces Free Bound.,} 2017) were obtained for essentially positively lower bounded kinetics or eventually which have some very weak degeneracy. Here we consider gene…

利用极大正则性求解一维燃烧型自由边界问题

Solutions to a One-Dimensional Combustion-Type Free Boundary Problem via Maximal Regularity

该研究针对燃烧理论中的一维自由边界问题，以极大$L^p$-$L^q$正则性为工具，建立了两种情形下解的存在唯一性、正则性及自由边界演化规律。

Interface regularity for semilinear one-phase problems

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. We prove a $C^{1,α}$ estimates for the "interfaces" (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole $\mathbb{R}^N$, for $N \leq 4$, answering positively a conjecture of Fernández-Real and Ros-Oton. Our results are to Bernoulli's free boundary problem what Savin's re…

Existence of a traveling wave solution in a free interface problem with fractional order kinetics

In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0 < $α$ < 1. We turn the free interface problem into a scalar free boundary problem coupled with an integral equation. The main intermediary step is to reduce the scalar problem to the study of a non-C 1 vector field in dimension 2. The latter is treated by qualitative topo-logical methods based on the Poincar{é}-Bendixson Theorem. The phase portrait is determined and the existence of a stable manifold at the…