Multiscale Analysis Modeling and Simulation Model Unit held intensive lectures (90 minutes x 4) by Professor Matthias Hieber (TU Darmstadt) from December 4 to 13, 2019. In this lecture, he explained the general theory of the maximal regularity theory for linear operators associated to the quasilinear parabolic type PDE. He also explained a unique existence theorem of local and global strong solutions and the stability of equilibria.
A quasilinear PDE is a PDE where a linear operator with a coefficient depending on unknown functions so that we can not apply the general theory of semilinear parabolic PDE due to Giga because we have a derivative loss with respect to time-variable when we rewrite the equation into the integral equation by using the Duhamel formula. To overcome this kind of difficulty, maximal regularity theory is adapted to construct a strong solution. Especially, this theory can be applied to several systems that appeared in physical phenomena including the free boundary problem of the Navier-Stokes equations that is one of the challenging problems nowadays. In fact, in this special lecture, Professor Hieber proved a local well-posedness of the system describing the behavior of the liquid crystal, the Ericksen-Leslie model, by using the abstract theory for maximal regularity theory. Furthermore, he showed a global well-posedness of the problem and some stability results for the equilibria.
Prof. Hieber is one of the great mathematicians who leads the study of maximal regularity theory for decades and has published numerous interesting research results. Some students did not major in mathematics, but thankfully for his kindness, he gave a lot of examples and explanations so that everyone could understand the lecture. This special lecture was very interesting and educational to learn the most advanced mathematical techniques.