A free-boundary problem in combustion theory
Summary: We consider the following problem arising in combustion theory: \[ \Delta u^\varepsilon- u^\varepsilon_t= v^\varepsilon f_\varepsilon(u^\varepsilon),\quad\Delta v^\varepsilon- v^\varepsilon_t= v^\varepsilon f_\varepsilon(u^\varepsilon)\quad\text{in }{\mathcal D}, \] where \({\mathcal D}\subset \mathbb{R}^{N+ 1}\), \(f_\varepsilon(s)= {1\over \varepsilon^2} f({s\over \varepsilon})\) with \(f\) a Lipschitz continuous function with support in \((-\infty, 1]\). Here \(v^\varepsilon\) is the mass fraction of some reactant, \(u^\varepsilon\) the rescaled temperature of the mixture and \(\varepsilon\) is essentially the inverse of the activation energy. This model is derived in the framework of the theory of equi-diffusional premixed flames for Lewis number 1. We prove that, under suitable assumptions on the functions \(u^\varepsilon\) and \(v^\varepsilon\), we can pass to the limit \((\varepsilon\to 0)\) -- the so-called high activation energy limit -- and that the limit function \(u= \lim u^\varepsilon= \lim v\varepsilon\) is a solution of the following free-boundary problem: \[ \Delta u- u_t= 0\quad\text{in }\{u> 0\},\quad |\nabla u|= \sqrt{2M(s, t)}\quad\text{on }\partial\{u> 0\}\tag{1} \] in a pointwise sense at regular free-boundary points and in a viscosity sense. Here \(M(x,t)= \int^1_{-w_0(x,t)}(s+ w_0(x,t))f(s) ds\) and \(-1< w_0= \lim_{\varepsilon\to 0}{v^\varepsilon- u^\varepsilon\over \varepsilon}\). Since \(v^\varepsilon- u^\varepsilon\) is a solution of the heat equation, it is fully determined by its initial-boundary datum. In particular, the free-boundary condition only (but strongly) depends on the approximation of the initial-boundary datum. Moreover, if \({\mathcal D}\cap\partial\{u> 0\}\) is a Lipschitz surface, \(u\) is a classical solution to (1).
scientific article; zbMATH DE number 1545561