Sebastian Schwarzacher

Department of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
Sokolovská 83
186 75 Praha 8

Email: schwarz@karlin.mff.cuni.cz

Tel:   (+420) 2 2191 3267

Curriculum Vitae (with list of publications)

Orcid (external)

Google Scholar (external)

MathSciNet (external)




Photo



Research interests/Scientific background

  • Nonlinear partial differential equations (existence, uniqueness, regularity, numerical analysis)
  • Fluid dynamics (Fluid-structure interactions, compressible fluids, non-Newtonian Fluids)
  • Calculus of variations (non-standard growth, rate independent systems, elastic solids)
  • Theory of Numerics for PDEs (time schemes, convergence rates, Galerkin methods)
  • Analysis of evolutionary non-linear PDEs (variable domains, intrinsic geometry, systems with variable contact interface)
Recently I specialized in the research on the mathematics for the interactions between (elastic) solids and (viscous) fluids. The research is supported by my working group. For more information on my working group please see the next paragraph.



Working group: Interaction of Fluids and Solids

The working group on the Interaction of Fluids and Solids was initiated at the beginning of 2019 and is mainly financed through third party fundings. For details please see the end of the paragraph. To stay in touch under the current measurements we have online team coffee-breaks every Monday and Thursday at 13.30. About every second week we have online seminar talks by group members. Anybody interested to participate can send me an email.

Research description

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 interchange ideas with the field of scientific computing and modeling and progress the related theory of numerical approximation schemes.

Current main scientific activities

  • A variational approach to fluid-structure interactions. Existence of weak solutions for 3D solids interacting with 3D fluids via DeGiorgi's celebrated minimizing movements method (Benesova, Kampschulte, Sch, 2020, in preparation).
  • Existence of weak solutions of unsteady fluids governed by the Navier-Stokes-Fourier equations interacting with elastic Koiter shells by combining ideas from the compressible existence (Breit, Sch, 2018, ARMA) with the developed regularity theory in (Muha, Sch, 2020, Preprint VI), (Breit, Sch, 2020, in preparation).
  • Numerical approximation of fluid-structure interaction. The development of (stable) schemes (Sch, She, 2020, Preprint II); the quantification of errors (motivated by the stability result shown in (Sch, Sroczinski, 2020, Preprint I) below); the study on adaptiv methods.
  • Bouncing of elastic objects in a fluid. We investigate the possibility of bouncing for solid objects hitting a wall in a viscous fluid. It is known that for no-slip boundary condition no contact of the solid with the wall is happening. We aim to find the necessary and sufficient elastic properties of the solid in order to bounce of the ground, even so no contact is happening.
    Numerical experiment: Bouncing of an elastic ball in a fluid with no-slip boundary conditions (Tuma, 2020).

Recent results

Results have been achieved on the weak-strong uniqueness (Sch, Sroczinski, 2020, Preprint I) of elastic plates interacting with incompressible fluids, on the development of a stable numeric approximation scheme for compressible fluids-structure interactions (Sch, She, 2020, Preprint II), as well as the existence and regularity of non-linear shells interacting with incompressible fluids in 3D (Muha, Sch, 2020, Preprint VI).

Team members:

Group meetings

We have regular group meetings every Thursday at 10.00 in the common room of the Department of Mathematical Analysis where results are discussed and presented. Everybody is welcome.

Funding

Funding for the project is provided by the support of the Junior Grant (GJ19-11707Y) of the Czech Science Foundation and by the Primus Research Programme (PRIMUS/19/SCI/01) of Charles University. Several members of the working group are additionally supported by the University Centre MathMAC (UNCE/SCI/023).

Poster (on the occasion of the Annual meeting of PRIMUS investigators 2020).


Photo

Photo



Teaching

Link to E-Lecture for Partial differential equations 2

Students that are interested in a Bachelor, Master or PhD thesis can find here a list of subjects.

Summerterm 2020:

Winterterm 2019/2020:

Summerterm 2019:

  • Mathematics 2 (for FSV UK). Tuesday 15:30 and Wednesday 11:00 in O 105. Find here the slides of the lecture.
    The written final exam is Tuesday 28.6.2019 in lecture hall K2 in Sokolovská 83 (2nd floor) from 14.00-15.30.

Winterterm 2018/2019:

  • Mathematics 1 (for FSV UK). Wednesday 9:30 and Thursday 11:00 in O 105. Find here the slides of the lecture. Find here the link to the web page of the related exercise classes.
    The written final exam is Monday 21.1.2019 in lecture hall O 105 from 9.30-11.00.
  • Mathematics 1 - Repetitorium (for FSV UK). Friday 9:30 in O 105.
  • Lecture K433KNM, Partial differential equations III. Monday at 14.00 in the seminar room of the KNM MFF UK (Karlin, 4th floor).

Summerterm 2017:

  • Mathematics 2 (for FSV UK). Tuesday 15:30 and Wednesday 12:30 in O 105.
  • Seminar on Differential Equations - NMMA431, Elliptic partial differential equations and free boundary problems. Anouncement.

Winterterm 2016/2017:

  • Analysis for instationary partial differential equations. Monday 10:40-12:20 in K9. Anouncement.



Recent Preprints

  1. S. Schwarzacher, M. Sroczinski: Weak-strong uniqueness for an elastic plate interacting with the Navier Stokes equation, (2020), Preprint.
  2. S. Schwarzacher, B. She: On numerical approximations to fluid-structure interactions involving compressible fluids, (2020), Preprint.
  3. C. Mindrila, S. Schwarzacher: Existence of steady very weak solutions to Navier-Stokes equations with non-Newtonian stress tensors, (2019), Preprint.
  4. O. Saari, S. Schwarzacher: A reverse Hölder inequality for the gradient of solutions to Trudinger's equation, (2019), Preprint.
  5. F. Rindler, S. Schwarzacher, J. J. L. Velazquez: Two-speed solutions to non-convex rate-independent systems, (2019), Preprint.
  6. B. Muha, S. Schwarzacher: Existence and regularity for weak solutions for a fluid interacting with a non-linear shell in 3D, (2019), Preprint.
  7. D. Breit, A. Cianchi, L. Diening, S. Schwarzacher: Global Schauder estimates for the p-Laplace system, (2019), Preprint.



Selected Publications

A complete list can be found in the Curriculum Vitae.

  • U. Gianazza and S. Schwarzacher: Self-improving property of the fast diffusion equation, JFA, Vol. 277, (2019). Preprint.
  • M. Bulicek, J. Burczak, S. Schwarzacher: Well posedness of nonlinear parabolic systems beyond duality, Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, (2019). Preprint.
  • U. Gianazza and S. Schwarzacher: Self-improving property of degenerate parabolic equations of porous medium-type, (2019), American Journal of Mathematics, Vol. 141, p. 399-446. Preprint.
  • Y. Lu and S. Schwarzacher: Homogenization of the compressible Navier-Stokes equations in domains with very tiny holes (2018), JDE 265 (4), 1371-1406. Preprint.
  • D. Breit and S. Schwarzacher: Compressible fluids interacting with a linear-elastic shell, ARMA, (2018), Vol. 228, p. 495-562. Preprint.
  • D. Breit, A. Cianchi, L. Diening, T. Kuusi and S. Schwarzacher: Pointwise Calderon-Zygmund gradient estimates for the p-Laplace system, JMPA, (2018), Vol.114, p. 146-190. Preprint.
  • F. Rindler, S. Schwarzacher, E. Suli: Regularity and approximation of strong solutions to rate-independent systems, M3AS, (2017), Vol.27, p. 2511-2556. Preprint.
  • L. Diening, S. Schwarzacher, B. Stroffolini and A. Verde: Parabolic Lipschitz truncation and Caloric Approximation, Calc. of Var. and PDE, (2017), Vol. 56, p. 120. Preprint.
  • M. Bulicek, J. Burczak and S. Schwarzacher: A unified theory for some non Newtonian fluids under singular forcing, SIAM J. Math. Anal., (2016), vol. 48, p. 4241–4267. Preprint.
  • M. Bulicek and S. Schwarzacher: Existence of very weak solutions to elliptic systems of p-Laplacian type, Calc. of Var. and PDE, (2016), vol. 55, 14 pp Preprint.
  • M. Bulicek, L. Diening and S. Schwarzacher: Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems, Analysis and PDE, (2016), vol. 9, p. 1115–1151 Preprint.
  • J. Frehse, S. Schwarzacher: On regularity of the time derivative for degenerate parabolic systems, SIAM J. Math. Anal., (2015), vol. 47, p.3917-3943 Preprint.
  • D. Breit, L. Diening, S. Schwarzacher: Finite element methods for the p(.)-Laplacian, SIAM J. Numer. Anal., (2015), vol. 53, p. 551-572 Preprint.
  • S. Schwarzacher: Hölder-Zygmund Estimates for Parabolic Degenerate Systems. J. Differential Equations, (2014), vol. 256, p. 2423-2448 Preprint.
  • D. Breit, L. Diening, S. Schwarzacher: Solenoidal Lipschitz truncation for parabolic PDE's. Math. Models Methods Appl. Sci., (2013), vol. 23, p. 2671-2700 Preprint.
  • L. Diening, Ch. Kreuzer, S. Schwarzacher: Convex Hull Property and Maximum Principles for Finite Element Minimizers of General Convex Functionals. Numer. Math. (2013), vol. 124, p. 685-700. Preprint.
  • L. Diening, P. Kaplicky, S. Schwarzacher: BMO estimates for the p-Laplacian, Nonlinear Anal., (2012), Vol. 75, p. 637-650, Preprint.