We prove that a suitable weak solutions of the three-dimensional MHD equations are smooth if the negative part of the pressure is suitably controlled.

Chair: Šárka Nečasová, Milan Pokorný

Yonsei University

Tuesday, 25. February 2020 - 9:00 to 10:00

We prove that a suitable weak solutions of the three-dimensional MHD equations are smooth if the negative part of the pressure is suitably controlled.

University of Florence

Tuesday, 25. February 2020 - 10:15 to 11:15

In crack nucleation and growth processes, the crack margins may be in contact although not linked by material bonds. In variational descriptions of fractures in elastic-brittle bodies -descriptions all based on energy minimization and De Giorgi's view on minimizing movements - the circumstance can be hardly described just by identifying cracks with the jump set of special bounded variation functions. On the other hand, if we consider both crack paths and deformations as distinct entities - although linked because the deformation jump set is at least included in the crack path - we face the basic problem of controlling minimizing sequences of crack surfaces in three-dimensional environment. The problem can be eluded if we can take - roughly speaking - just sequences of surfaces with bounded curvature, although we should think of the notion of curvature we use because crack surfaces could be extremely rough. To this aim,... more