In between elliptic PDEs, which do not depend on time (think of the steady-state Stokes equations), and honest parabolic PDEs, which do depend on time and are started at a given initial value (think of the instationary Stokes equations), there are time-periodic parabolic PDEs: On the one hand, time-independent solutions to the elliptic PDE are also trivially time-periodic, which gives periodic problems an elliptic touch, on the other hand solutions to the initial value problem which are not constant in time might very well be periodic.
I want to advocate for time-periodic problems not being the little sister of either elliptic or parabolic problems, but being a connector between the two and a class of its own right. This is highlighted by a direct method for showing a priori Lp estimates for time-periodic, linear, partial differential equations. The method is generic and can be applied to a wide range of problems, for example the Stokes equations and boundary value problems of Agmon-Douglas-Nirenberg type. In the talk, I will present these ideas and show how they can be extended to the setting of weighted Lp estimates, which is advantageous for extrapolation techniques and rougher boundary data.
Parts of the talk are based on joint works with Yasunori Maekawa and Mads Kyed.
This video was produced by the SITE Research Center at New York University, as part of their talk series.
