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

Some results on compressible magnetohydrodynamic system with large initial data

Yang Li
Institute of Mathematics, CAS
Tuesday, 26. March 2019 - 9:00 to 10:00
The time evolution of electrically conducting compressible flows under the mutual interactions with the magnetic field is described by the system of magnetohydrodynamics (MHD). In this talk, we focus on the existence of global-in-time weak solutions with large initial data. Firstly, for a simplified 2D MHD model of viscous non-resistive flows, we prove the existence of global-in-time weak solutions by invoking the idea from compressible two-fluid model. Secondly, for the general 3D inviscid resistive MHD system, we prove the existence of infinitely many global-in-time weak solutions for any smooth initial data. To do this, we appeal to the method of convex integration developed by De Lellis and Szekelyhidi and adapted to the compressible flows by Chiodaroli, Feireisl and Kreml.
The results are based on the joint works with Eduard Feireisl and Yongzhong Sun.

The incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids

Aneta Wroblewska-Kamińska
Institute of Mathematics, Polish Academy of Sciences
Tuesday, 19. March 2019 - 9:00 to 10:00
We explore the behaviour of global-in-time weak solutions to a class of bead-spring chain models, with finitely extensible nonlinear elastic (FENE) spring potentials, for dilute polymeric fluids. In the models under consideration the solvent is assumed to be a compressible, isentropic, viscous, isothermal Newtonian fluid, confined to a bounded open domain in R^3, and the velocity field is assumed to satisfy a complete slip boundary condition. We show that for ill-prepared initial data, as the Mach number tends to zero, the system is driven to its incompressible counterpart.
The result is a joint work with Endre Süli.

Wild solutions to isentropic Euler equations starting from smooth initial data

Ondřej Kreml
Institute of Mathematics, CAS
Tuesday, 12. March 2019 - 9:00 to 10:00
We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of smooth initial data which admit infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant T > 0. In order to continue the solution after the formation of the discontinuity, we apply the theory developed by De Lellis and Szekelyhidi and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution.
This is a joint work with Elisabetta Chiodaroli, Václav Mácha and Sebastian Schwarzacher.