题目：An Introduction to Stability of Traveling Wave Solutions in Free Interface Problems
报告人：Claude-Michel Brauner (Tongji and Bordeaux (France) Univ.)
时间：2020-11-19, 12-03, 12-17（星期四）上午8：00—10：00
In this round of talks, we intend to discuss a method especially suitable for the analysis of stability of traveling waves in free interface problems. For convenience, some examples are taken from combustion theory, however our approach is quite general and may apply to a whole gamut of problems. As usual, the stability issue includes the study of perturbations of the traveling wave and the analysis of the linearized operator in appropriate functional spaces. Thanks to an ansatz, we show how the free interface can be eliminated. Moreover, we find a second-order Stefan condition for the velocity of the interface. We eventually derive an equivalent fully nonlinear problem for which the linearized stability principle holds. The theory of analytic semigroups plays a crucial role.
November 19: In the first talk, we present the basic feature of the method in the case of the Arrhenius kinetics. At the free interface, namely the flame front, the solution is continuous, however the gradient is discontinuous and the jump is given. In contrast to classical Stefan problems, there is no condition for the interface velocity.
December 3: We generalize the previous analysis to the abstract setting of parabolic overdetermined problems in the sense of Hadamard. In particular, we define a transversality (or non-degeneracy) condition: a free interface is said non-degenerated if this condition is verified.
December 17: We consider a model of premixed flame propagation characterized by two free interfaces, respectively the ignition and the trailing fronts, based on an ignition-temperature kinetics. In contrast to Arrhenius kinetics, the gradient is continuous at the interfaces. It appears that the ignition interface is non-degenerated, while the trailing interface is degenerated because the gradient vanishes. To address this difficulty, a trick is to differentiate the system. This problem has a different sort of complexity as 2D perturbations of the traveling wave generate instability and cellular structure.