On shifting the thermal explosion threshold by a vortical flow in dimension two
This paper is concerned with a study of a natural generalization of a classical Frank-Kamenetskii model of thermal explosion in the presence of a vortical flow in a two dimensional setting. This model describes possible stationary temperature distributions in a combustion vessel which boundary is maintained at a constant temperature. The model constitutes a Dirichlet boundary value problem for a certain semi-linear elliptic equation that depends on a parameter $\lambda,$ called Frank-Kamenetskii parameter. A remarkable property of this problem is that it admits a classical minimal solution when the Frank-Kamenetskii parameter does not exceed some critical value $\lambda^*$ and no classical solutions for $\lambda>\lambda^*$. The absence of a classical solution, in the framework of Frank-Kamenetskii theory, is associated with the thermal explosion event. Consequently, in the context of combustion, $\lambda^*,$ commonly called an explosion threshold, is a maximal value of the Frank-Kamenetskii parameter which allows to attain a thermal equilibrium within a combustion vessel and thus provides a sharp characterization of the thermal explosion. A critical temperature distribution corresponding to $\lambda^*$ is called an extremal solution. In this paper, we show that, under an assumption of sufficiently fast growth of the reaction term, there exists a regular vortical flow that allows to adjust an explosion threshold by reversing its direction, provided a combustion vessel is not a disk. We also give rather detailed description of extremal solutions. In particular, we show that extremal solutions are always classical.