## 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 study the time evolution of non-linear viscoelastic solids in the presence of inertia and (self-)contact. For this problem we prove the existence of weak solutions for arbitrary times and initial data, thereby solving an open problem in the field. Our construction directly includes the physically correct, measure-valued contact forces and thus obeys conservation of momentum and an energy balance. In particular, we prove an independently useful compactness result for contact forces.

#### Stability and convergence of in time approximations of hyperbolic elastodynamics via stepwise minimization (with S. Schwarzacher)

Journal of Differential Equations, 415:434 486 (2025)

arXiv:2305.19880

We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics, namely so-called generalized solids, undergoing large deformations. The evolution follows an underlying variational structure exploited by step-wise minimization. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can be assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.

#### Inertial (self-)collisions of viscoelastic solids with Lipschitz boundaries (with G. Gravina and M. Kampschulte)

Advances in Calculus of Variations, 2024 (to appear)

arXiv:2312.00431

We continue our study, started in a previous work, 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

Nonlinear Analysis, 245:113554 (2024)

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)

## Teaching - present

Currently I am not teaching.