Weak Solutions to Mathematical Models of the Interaction between Fluids, Solids and Electromagnetic Fields

• Název práce v češtině: Slabé řešení matematických modelů pro interakci mezi tekutinami, pevnými látky a elektromagnetickým polem
• Název v anglickém jazyce: Weak Solutions to Mathematical Models of the Interaction between Fluids, Solids and Electromagnetic Fields
• Klíčová slova: Interakce tekutin a pevných látek|magneto-elasticita|magneto-hydrodynamika|minimalizujíci posuny|Navierovy-Stokesovy rovnice|Rotheho metoda
• Klíčová slova anglicky: Fluid-structure interaction|Magnetoelasticity|Magnetohydrodynamics|Minimizing movements|Navier-Stokes equations|Rothe method
• Akademický rok vypsání: 2019/2020
• Typ práce: disertační práce
• Jazyk práce: angličtina
• Ústav: Katedra matematické analýzy (32-KMA)
• Vedoucí / školitel: Mgr. Barbora Benešová, Ph.D.
• Řešitel: skrytý - zadáno a potvrzeno stud. odd.
• Datum přihlášení: 16.07.2019
• Datum zadání: 16.07.2019
• Datum potvrzení stud. oddělením: 04.10.2019
• Datum a čas obhajoby: 23.01.2024 14:00
• Datum odevzdání elektronické podoby: 15.10.2023
• Datum odevzdání tištěné podoby: 27.10.2023
• Datum proběhlé obhajoby: 23.01.2024
• Oponenti: prof. Boris Muha
• Konzultanti: RNDr. Šárka Nečasová, DSc.

Zásady pro vypracování

The key theme of this thesis is the mathematical analysis, i.e. the proof of existence, uniqueness and regularity of solutions, for models describing fluids, solids or their interactions that show complex behavior. These models are either phrased as systems of partial differential equations or as variational problems of energy functionals. The research in this thesis is guided by the long-term goal to be able to provide well-possedness for models of fluid-structure interaction of a general (e.g. compressible, heat-conducting) fluid with a non-linear solid that can undergo large deformations. Such interactions appear in many research areas which range from biomedicine (blood flow in vessels, artificial heart valves, windpipes, airway closure in lungs) to geophysics (underground flows, hydraulic fracturing, dynamics of magnetic intrusions ) and industrial applications (aero-elasticity, offshore structures, etc.)

To reach the long-term goal outlined above, further insight into the analysis of models describing fluids, and solids as well as the study of simplified situations in fluid-structure interaction are necessary. Within this thesis, among the possible research directions we propose studying the fluid-structure interaction of a quite general fluid (compressible, magnetic, heat-conducting) with a very simplified behavior of the solid structure (e.g. rigid body, simple elastic-shell structure). Here, the aim is to generalize works [Fe], [CMN], [BrS]. Another direction is the study of the solid material alone, in order to assure injectivity of the obtained solutions, which is needed in fluid-structure interaction problems. This is a notoriously difficult problem even in the sationary case (see [Ball02]), however more information can be obtained if higher-gradient regularizations are included into the model (e.g. [BKS]). Studying these in the evolutionary case seems to be a prospective way to obtain analytical results in elastodynamics, where only a few results are available (e.g. [DST1], [DST2]) and could eventually be used in fluid-structure interaction.

This thesis is perfomed in cooperation ("cotutelle de these") between Charles University and University of Würzburg (guided by Prof. Dr. Anja Schlömerkemper). On the Prague side it is supported by the PRIMUS and GAČR (GA19-U707Y) grants of S. Schwarzacher and is thus embedded into reserach groups on fluid-structure interactions both at the Department of Mathematical Analysis, Faculty of Mathematics and Physics at Charles Univeristy as well as the Mathematical Institute of the Czech Academy of Sciences.

Seznam odborné literatury

[Ball02] Ball, J. M. Some open problems in elasticity. In Geometry, mechanics, and dynamics (pp. 3-59). Springer New York, 2002.

[BK17] B. Benešová , M. Kružík: Weak lower semicontinuity of integral functionals and applications. SIAM Review, 59(4) . (2017)., 703-766.

[BKS] B. Benešová, M. Kružík, A. Schlömerkemper: A note on locking materials and gradient polyconvexity, Math. Mod. Meth. Appl. Sci. 28 (2018), 2367–2401.

[BrS] D. Breit and S. Schwarzacher: Compressible fluids interacting with a linear-elastic shell, ARMA, (2018), Vol. 228, p. 495-562

[CMN] N. V. Chemetov, Š. Nečasová, B. Muha: Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition. J. Math. Phys. 60 (2019), no. 1, 011505, 13 pp.

[DST1] S. Demoulini, D. M. Stuart, and A. E. Tzavaras: A Variational Approximation Scheme for Three-Dimensional Elastodynamics with Polyconvex Energy, ARMA 157.4 (2001): 325-344.

[DST2] S. Demoulini, D. M. Stuart, and A. E. Tzavaras: Weak–strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics.: ARMA 205.3 (2012): 927-961.

[Fe] E. Feireisl: On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167 (2003), no. 4, 281–308.