Entropy methods for higher-order parabolic equations

In the context of parabolic evolution equation, the entropy method is an analytical machinery that uses Lyapunov functionals to establish the existence of weak solutions and to extract relatively precise information on their speed of convergence to equilibrium. In this talk, we first recall the basic ideas behind the entropy method by reviewing the classical results on self-similar asymptotics for second order porous medium equations. Then we apply the machinery to a family of degenerate fourth order equations from the context of thin film theory. Finally, we discuss the entropy approach to a degenerate equation of sixth order, which is related to a diffusive model for quantum systems. In both situations, we are able to prove the existence of global weak solutions (that are classical solutions "locally in time"), and to calculate a bound on the rate of equilibration.

The part on fourth order equations is a collaborative work with Giuseppe Savare and Stefano Lisini from Pavia; the part on sixth order equation is joint work with Ansgar Juengel and Mario Bukal from Vienna.