The Centre of analysis for the interaction of fluids and solids was initiated at the beginning of 2019. From 09/2021 until 08/2026 it was mainly financed through the ERC-CZ Grant LL2105 CONTACT (at Charles University, Prague) and the VR Grant 2022-03862 (at Uppsala University). Currently the team has 9 members.
The research is focused on the interactions between fluids and solids. Fluid solid interaction happens in many everyday instances. For example blood flow through a vessel or air flow through the trachea, oscillations of suspension bridges, lifting of airplanes, bouncing of elastic balls, or the rotation of wind turbines. The working group aims to systematically develop an analysis for the related theory of partial differential equations. We attack classical questions of existence, uniqueness, regularity and stability, questions about the qualitative behavior of fluids interacting with solids and the quantification of the forces at the free interface between the solid and the fluid – the variable domain. Moreover, we progress in the field of scientific computing and modeling. In case you wish to participate in the program please contact S. Schwarzacher for more information.
The ERC-CZ project CONTACT (2021–2026)
The CONTACT project advanced mathematical theory and numerical methods for fluid-structure interaction. It delivered new results on existence, regularity and contact, explained the mechanism of contactless rebound, and developed numerical methods and an international benchmark for its simulation. It opened new directions in time-periodic and variational methods and built an international network.
See the list of publications and the list of conferences and talks.
Milestone results
- B. Benešová, M. Kampschulte, S. Schwarzacher: A variational approach to hyperbolic evolutions and fluid-structure interactions, JEMS 26 (2024). A new variational existence scheme for largely deforming viscoelastic solids interacting with the unsteady Navier-Stokes equations.
- G. Gravina, S. Schwarzacher, O. Souček, K. Tůma: Contactless rebound of elastic bodies in a viscous incompressible fluid, JFM 942 (2022). The flattening of the body close to the wall is essential for a physical rebound.
- M. Kampschulte, S. Schwarzacher, G. Sperone: Unrestricted deformations of thin elastic structures interacting with fluids, JMPA 173 (2023). Includes the universal Bogovskij operator, a tool used in several further works.
- S. Schwarzacher, P. Su: Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations, CMP 406 (2025). The dissipation of the fluid alone yields strong solutions.
- S. Mosny, B. Muha, S. Schwarzacher, J. T. Webster: Time-periodic solutions for hyperbolic-parabolic systems, accepted in JEMS (2026), arXiv:2412.18801. New a-priori estimates controlling waves through a dissipative interface.
Simulation from the benchmark for fluid-structure-contact interaction. More simulations and videos.
Research
Elastic plates and shells interacting with fluids
Strong solutions for a perfectly elastic beam interacting with the 2D Navier-Stokes equations were constructed for large data and long times (Schwarzacher, Su, 2025, CMP). For 3D fluids a Ladyzhenskaya-Prodi-Serrin condition was shown (Breit, Mensah, Schwarzacher, Su, 2025, Ann. SNS Pisa) and well-posedness for perfectly elastic shells in (Breit, Mensah, Schwarzacher, Su, 2026). These results will be collected in a monograph. Weak solutions involving displacements in all coordinate directions were constructed by the variational method (Kampschulte, Schwarzacher, Sperone, 2023, JMPA); Bogovskij operators in non-cylindrical domains are given in (Saari, Schwarzacher, 2023, Ann. PDE). Heat conducting fluids interacting with elastic shells are treated in (Breit, Schwarzacher, 2023, Ann. SNS Pisa) and (Mitra, Schwarzacher, 2025). With P. Su and M. Hillairet contactless rebounds of viscoelastic plates in a fluid are studied.
Fluid-structure interaction with bulk solids
Existence of weak solutions for 3D solids interacting with 3D fluids via De Giorgi's minimizing movements method (Benešová, Kampschulte, Schwarzacher, 2024, JEMS), see also the survey (Benešová, Kampschulte, Schwarzacher, 2023, Nonlin. Anal. RWA). The method was extended to compressible fluids (Breit, Kampschulte, Schwarzacher, 2024, Math. Ann.) and to slip boundary conditions (Češík, Kampschulte, Schwarzacher, 2026). Contact and collisions of elastic solids were studied in (Češík, Gravina, Kampschulte, 2024, Calc. Var. PDE) and (Kampschulte, Muha, Trifunović, 2024, JDE), time discretisation in (Češík, Schwarzacher, 2025, JDE). Regularity for largely deforming solids with temperature effects is shown in (Almi, Badal, Friedrich, Schwarzacher, 2026, Calc. Var. PDE).
Contact and forces on solids in fluids
Elastic bodies can rebound in a viscous fluid without topological contact; the flattening of the body close to the wall is essential (Gravina, Schwarzacher, Souček, Tůma, 2022, JFM). Re-meshing strategies for the simulation of such rebounds were developed in (Fara, Schwarzacher, Tůma, 2024, CMAME), which led to the international benchmark (Corti, Fara, Fernández, Frei, Knoke, Schwarzacher, Tůma, Wick, 2026). For an obstacle without a nearby wall convection is essential: drag and lift change significantly with the Reynolds number (Cach, Tůma, Blechta, Schwarzacher, 2025). A linear numerical scheme for fluid-structure interaction with stability and error estimates is given in (Schwarzacher, She, Tůma, 2025, Numer. Math.). See Numerics.
Time-periodic solutions and damping
The damping effect of a fluid on a solid leads to the study of coupled heat-wave systems. Existence of time-periodic solutions under geometric conditions was shown in (Mosny, Muha, Schwarzacher, Webster, accepted in JEMS); that these conditions are qualitatively sharp and how they relate to resonance is shown in (Muha, Schwarzacher, Webster, 2026). Time-periodic solutions for shells: (Mîndrilă, Schwarzacher, 2023).
New directions
- Elasto-plastic solids interacting with fluids (T. Biswas, B. Benešová, S. Schwarzacher): existence, regularity and uniqueness for small strain elasto-plasticity.
- Variational numerical methods for large strain elasticity (M. Fehling, K. Tůma, S. Hencl, S. Schwarzacher). M. Fehling is one of the principal developers of the open-source finite element library deal.II.
- Parabolic problems in time-dependent domains (D. Campbell, L. Schätzler, S. Schwarzacher): trace and extension theorems.
- Free surface flows (J. Niinikoski, M. Kampschulte, S. Schwarzacher): higher order estimates (2024) and varifold solutions.
Funding
The Ministry of Education, Youth and Sport of the Czech Republic (MSMT) supported the centre via the ERC-CZ Grant LL2105 CONTACT from 09/2021 until 08/2026. Further we thank the support of the University Centre MathMAC (UNCE/SCI/023) and (UNCE/24/SCI/005), and the Swedish research council via the VR Grant 2022-03862.
