Antonín Češík
PhD student at Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University.
Working as a member of research group on Interaction of Fluids and Solids, which is led by my advisor Sebastian Schwarzacher .
Research
Inertial evolution of (visco-)elastic solids with collisions (with G. Gravina and M. Kampschulte)
Calculus of Variations and PDE 63(2):55 (2024). arXiv:2212.00705
We extend the existence results for dynamics of (visco-)elastic solids past the time of a (self-)collision. In particular, we assume only the non-interpenetrability of the solid, and our construciton directly provides the physically correct contact forces. In particular, the contact force constructed as a Lagrange multiplier in fact turns out to be vector-valued measure, supported at the points of contact and normal to the boundary.
Analysis of a variational approach to hyperbolic problems (with S. Schwarzacher)
arXiv:2305.19880
The aim is to investigate the properties of a variational time-discrete scheme for solving hyperbolic problems, in particular this includes dynamic evolutions of (visco-)elastic solids. The scheme is now fully discrete in time, satisfies the right energy estimates. Moreover we show the rate of convergence of this scheme to the continuous (strong) solution, given sufficient regularity.
Inertial (self-)collisions of viscoelastic solids with Lipschitz boundaries (with G. Gravina and M. Kampschulte)
arXiv:2312.00431
We continue our study, started in arXiv:2212.00705, of (self-)collisions of viscoelastic solids in an inertial regime. We show existence of weak solutions with a corresponding contact force measure in the case of solids with only Lipschitz-regular boundaries. This necessitates a careful study of different concepts of tangent and normal cones and the role these play both in the proofs and in the formulation of the problem itself. Consistent with our previous approach, we study contact without resorting to penalization, i.e. by only relying on a strict non-interpenetration condition. Additionally, we improve the strategies of our previous proof, eliminating the need for regularization terms across all levels of approximation.
Convex hull property for elliptic and parabolic systems of PDE
arXiv:2311.16949
We study the convex hull property for systems of partial differential equations. This is a generalisation of the maximum principle for a single equation. We show that the convex hull property holds for a class of elliptic and parabolic systems of non-linear partial differential equations. In particular, this includes the case of the parabolic p-Laplace system. The coupling conditions for coefficients are demonstrated to be optimal by means of respective counterexamples.
Convex hull properties for parabolic systems of PDE (master thesis)
The thesis investigates a generalisation of the maximum principles for PDE into the context of systems of PDE - the convex hull property. The main novelty is a proof of convex hull property for the parabolic p-Laplacian.
Teaching - present
Currently I am not teaching.