Speaker: Hussain Ali Ibdah
Please note that this seminar will take place online via Zoom. You can connect to this seminar via the following link:
Abstract: In this talk, I will explain how to obtain a global well-posedness and regularity result of strong solutions to a slight modification of the so called Michelson-Sivashinsky equation. This equation is a special case of the classical combustion model derived by Sivashinsky, more widely known as the Kuramoto-Sivashinsky equation. The difference is that the dissipative operator is the standard Laplacian while the instabilities manifest themselves in the form of the fractional Laplacian, as opposed to bi-Laplacian vs. Laplacian in the Kuramoto-Sivashinksy equation. The difficulty arises in dimensions higher than one due to the lack of any obvious a-priori estimates. Regularity is shown to persist by adopting ideas introduced by Kiselev, Nazarov, Volberg and Shterenberg to handle the critically dissipative SQG and Burgers equation. Namely, the Lipschitz constant of the solution is shown to be under control by constructing a modulus of continuity that must be obeyed by the solution. If time permits, I will briefly explain how this equation can serve as a toy model for the incompressible Navier-Stokes system and discuss current work in progress in which an attempt to extend those ideas to the latter case is being studied.