Seminar

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract :

In 1975, G.Da Prato and P. Grisvard have proved a generic Lq-Maximal Regularity theorem which provides a Lq-maximal regularity on appropriate real interpolation spaces associated to the generator of the involved semi-group. In November 2020, Danchin, Mucha, Hieber and Tolksdorf gave a more general result implying homogeneous estimates, which provide global time control.

The goal here is to apply the last result to the Hodge Laplacian defined on 1-forms on Rd+, noticing that in this case one can fully diagonalize the latter as Dirichlet and Neumann Laplacians acting separately on each component.

So, for appropriate definitions of Homogeneous Sobolev and Besov Spaces, it will be sufficient to give a full description of the real interpolation space between Lp and the Homogeneous Sobolev spaces with Dirichlet or Neumann boundary condition, which will be actually the Homogeneous Besov Space with appropriate Boundary Condition. The result will follow by our "diagonalization" and the 2020 - Da Prato-Grisvard theorem leading to global time estimates for the Hodge Laplacian with homogeneous Boundary condition.

If we have enough time we are going to briefly discuss the associated Hodge-Stokes problem.

Graduate Student Working Group