slideshow 3

Seminar on Partial Differential Equations

usually takes place each Tuesday at 09:00 in IM, rear building, ground floor
Chair: Šárka Nečasová, Milan Pokorný

Up to the boundary Lipschitz regularity for variational problems

Erika Maringová
Charles University
Tuesday, 6. March 2018 - 9:00 to 10:00
We prove the existence of a regular solution to a wide class of convex, variational integrals. Via technique of construction of the barriers we show that the  solution is Lipschitz up to the boundary. For the linear growth case, we identify the necessary and sufficient condition to existence of solution; in the case of superlinear growth, we provide the sufficient one. The result is not restricted to any geometrical assumption on the domain, only its regularity plays the role.

Weak solutions to a two-phase thin film equation with insoluble surfactant

Gabrielle Brüll
Norwegian University of Science and Technology
Tuesday, 27. February 2018 - 9:00 to 10:00
We discuss a model describing the spreading of an insoluble surfactant on the upper surface of a viscous complete wetting two-phase thin film. Considering capillary effects as the only driving force, the system of evolution equations consists of two strongly coupled degenerated equations of fourth order describing the film heights of the fluids, which are additionally coupled to a second-order transport equation for the surfactant concentration. Owing to the degeneracy, it is in general not clear whether one can prove the existence of global solutions in a classical sense, which motivates the study of weak solutions. The proof of existence of nonnegative global weak solutions is based on a priori energy estimates and compactness arguments.

Regularity and uniqueness for a critical Ladyzhenskaya fluid

Dalibor Pražák
Charles University
Tuesday, 20. February 2018 - 9:00 to 10:00
We consider an incompressible p-law type fluid in a 3D bounded domain. Employing iterative estimate in Nikolskii spaces and reverse Hölder inequality, we establish higher time regularity and uniqueness of weak solution provided the data are more regular.
This is a joint work with M. Bulíček and P. Kaplický.

Some properties of the strong solution of the Navier-Stokes equations

Petr Kučera
Czech Technical University
Tuesday, 9. January 2018 - 9:00 to 10:00