Parabolic quasilinear evolution equations naturally occur in mathematical fluid dynamics of complexly coupled fluid systems. One striking example are the Ericksen-Leslie equations modeling the dynamics of nematic liquid crystals – a material having both the characteristics of a fluid, namely a flow property, and also possessing structural properties of a crystal, namely a molecular orientational order.
In her thesis, Katharina Schade applies modern parabolic quasilinear theory to several systems related to the Ericksen-Leslie theory and arrives at a comprehensive understanding from the point of view of dynamical systems. These systems include the simplified Ericksen-Leslie equations (Lin-Liu 1995), thermodynamic and compressible extensions as well as a parameter-restricted version of the full Ericksen-Leslie equations.
In parabolic theory, understanding underlying linear problems is key for understanding non-linear systems. The author considers the notorious case of Lebesgue index p=∞ for the Lp-Stokes problem.
The question whether the Stokes operator generates an analytic semigroup in a space of essentially bounded solenoidal functions in cylindrical domains, is answered affirmatively using a de Giorgi-type contradiction argument.