
Mathematical theory and numerical analysis for equations of viscous newtonian compressible fluids (GA2201591S)
from 01/01/2022
to 31/12/2024 main investigator

Objectives:
Equations of compressible viscous fluids are important models in many applications. We will study the corresponding systems of partial differential equations from several points of view: existence theory and qualitative properties of solutions for different choices of boundary conditions (including open system), different types of domains (in particular varying in time), different types of solutions (weak, strong, dissipative) and different simplified models (in
particular, compressible primitive equations) as well as from the point of view of numerical mathematics (construction of benchmarks, numerical analysis of some methods, comparision of different numerical methods). The proposal of the project is based on a close collaboration of specialists from different mathematical disciplines.



Qualitative Theory of the MHD and Related Equations (GC2208633J)
from 01/01/2022
to 31/12/2024 investigator

Objectives:
The project is oriented to studies of a wide class of equations of magnetohydrodynamics (MHD), modeling flows of electrically conductive fluids (both incompressible and compressible, without or with Hall's effect). The presence of magnetic field evokes many new challenging questions. In addition to existing mathematical models, our aims include formulation and studies of new models, which e.g. concern (a) physically and mathematically relevant boundary conditions for the magnetic field, (b) micropolar fluids and (c) motion of a body with a cavity filled in by an electrically conductive fluid. Considered mathematical problems involve questions of regularity, analysis and consistent theory for the models from items (a)(c), problems with artificial "outflow"' boundary conditions and problems in domains with moving boundaries. Our project assumes a close cooperation with colleagues from the Yonsei University in Seoul and other respected researchers and active participation of postdocs and doctoral students on both sides.



Problémy interakce tekutiny se strukturou: matematická analýza a aplikace (Praemium Academiae  ŠN)
from 01/01/2022
to 31/12/2027 main investigator

Objectives:
Hlavní náplní projektu Š. Nečasové je výzkum zaměřený na matematickou analýzu modelů mechaniky tekutin (primárně stlačitelných) a dynamiky pevných látek, včetně jejich interakce na společné hranici. Problémy tohoto typu se přirozeně vyskytují v aplikacích v průmyslu, biomedicíně (proudění krve) a ve vědách o životním prostředí (oceánografie, meteorologie).



Partial differential equations in mechanics and thermodynamics of fluids (GA1904243S)
from 01/01/2019
to 31/12/2021 main investigator

Objectives:
Partial differential equations in mechanics and thermodynamics of fluids are a mathematical tool which models the time evolution of basic physical quantities. The goal of this project is to study these systems of partial differential equations from the point of view of mathematical and numerical analysis and to compare these results with the properties of their numerical solutions. We will mostly deal with solvability of the problems (existence of solutions for different formulations, possibly their uniqueness), qualitative properties of the solutions, analysis of the adequate numerical methods and numerical solutions of these problems. The proposal of the project assumes a close collaboration of specialists from different branches. Such a collaboration stimulates positively the developement of all participating mathematical disciplines.



Flow of viscous fluid in time dependent domain (7AMB16PL060)
from 01/01/2017
to 31/12/2018 main investigator

Objectives:
G1. Global existence of weak solution of full system in timedependent domain.
Goals: Our goals is the study of the global existence of a weak solution in the case of a bounded or an exterior domains for the Navier or Dirichlet type of boundary conditions. We will use the method introduced in E. Feireisl: Dynamics of viscous compressible fluids, 2004 for fixed domain and we will also apply the penalization method.
G2: Relative entropy inequality.
Goals: We will focus on the derivation of the relative entropy inequality in the case of timedependent domain. We will use the results from fixed domain introduced in E. Feireisl and A. Novotný: Weakstrong uniqueness for the full NavierStokesFourier system, 2012.
G3. Singular limits.
Goals We shall study the singular limits in the regime of low Mach number. From this follows that the limit system (target system) is the system of incompressible flow in the timedependent domain. We will use the results from E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa, J. Stebel: Incompressible limits of fluids excited by moving boundaries, 2014, where barotropic case was studied, and also the results for fixed domains.
G4: Weak solution of viscous flow around rotating rigid body and his asymptotic behavior.
Goals: We will focus on the problem of existence of weak solution in weighted Lorentz spaces to get the asymptotic behavior of flow around a body. Because of the lack of the regularity in Lp spaces it is necessary to go into more complicated structure of Lorentz spaces, where the integrability of convective term is satisfied. Moreover, we would like to study the relative entropy inequality in the case of compressible flow around rotating body.



Theory and numerical analysis of coupled problems in fluid dynamics (GA1701747S)
from 01/01/2017
to 31/12/2019 investigator

Objectives:
The project is focused on several important fields of today's rapidly developing mathematical fluid mechanics. The aim is to derive a series of results, from new regularity criteria, stability and robustness analysis of solutions, up to the low Mach and high Reylolds limits in a compressible fluid interacting with a solid structure. Beside the qualitative analysis of flow problems, a part of the project is the development and analysis of new, accurate and robust numerical methods for the solution of important and topical models of fluid dynamics. The attention will be paid to the development and analysis of high order methods for the solution of nonstationary nonlinear partial differential equations and compressible flow, based on the discontinuous Galerkin method. Particularly we have hpversions in mind. These methods will be applied to the numerical solution of fluidstructure interaction and multiphase flow. Another subject is the study of flow model with slip boundary conditions.



Thermodynamically consistent models for fluid flows: mathematical theory and numerical solution (GA1603230S)
from 01/01/2016
to 31/12/2018 main investigator

Objectives:
Mathematical and numerical analysis and numerical solution of fluid flows belong to the most often studied problems of the theory of partial differential equations and their numerical solution. During the last decades, a big progress has been achieved in these fields which enables us to study models of complex fluids including the possibility to consider their dependence on temperature. This project is focused on the study of such models of fluid thermodynamics and mechanics with the aim to extend the knowledge in the field of the theoretical analysis of the corresponding systems of partial differential equations and numerical analysis of the methods for their solution. Computational simulations using specific numerical methods will be performed to support the analytical results concerning the wellposedness of the model problems and qualitative properties of their solutions. The proposed projects assumes a tight collaboration of specialists in these fields which is an important prerequisite for further development of mathematical and computational fluid thermodynamics.



Qualitative analysis and numerical solution of problems of flows in generally timedependent domains with various boundary conditions (GA1300522S)
from 01/02/2013
to 31/12/2016 investigator

Objectives:
Mathematical and computational fluid dynamics play an important role in many areas of science and technology. The project will be concerned with the analysis of qualitative properties of the incompressible and compressible NavierStokes equations in fixed or timedependent domains with various types of, in general nonstandard, boundary conditions. Let us mention, e.g., the existence, uniqueness, regularity and singular limits of their solutions. On the basis of theoretical results, in the numerical part of the project, we shall develop efficient and robust techniques for the solution and validation of theoretically analyzed flow problems and models. The developed numerical methods and their ingredients, as, e.g., adaptivity and hpmethods, will be tested on suitable problems and applied to fluidstructure interaction. With the aid of model problems, theoretical aspects of the worked out methods as stability, convergence and error estimates will be investigated.



Flow of fluids in domains with variable geometry (P201/11/1304)
from 01/01/2011
to 31/12/2013 main investigator

Objectives:
The goal of the project is to get new relevant results concerning flow in domains with varying geometry. From the viewpoint of theoretical analysis, we will deal with flow of fluids (incompressible and compressible) around a rotating body (existence of weak or very weak solutions, asymptotic behaviour solutions, artificial boundary conditions) in case that the axis of rotation of the body and the velocity at infinity are parallel or not parallel. We will also investigate the related hydrodynamical potential theory. Moreover, we will investigate the case of motion of rigid bodies in viscous fluid (mostrly nonNewtonian incompressible and Newtonian compressible), in several cases we include the changes of temperature. Part of the problems mentioned above will be solved numerically. Finally, we perform the numerical simulation of flow of fluids in domains with complicated geometry corresponding to the flow of blood in healthy veins as well as in cases of cardiovascular diseases.



Contemporary function spaces theory and applications (P201/10/1920)
from 01/01/2010
to 31/12/2012 main investigator

Objectives:
This project can be briefly characterized as investigation of imbedding and trace properties of weighted and anisotropic spaces of Sobolev type, research in the extrapolation theory, and applications to the qualitative theory of differential equations, particularly to the Stokes and Oseen problem and also to the NavierStokes systems. Specifically we intend to study imbeddings, traces ad interpolation inequalities in general spaces of Besov and LizorkinTriebel type with dominating mixed smoothness in the framework of the Fourier analytic approach to the theory, and some of the related unsolved problems not falling into this general scheme as reduced imbeddings and inequalities in Orlicz spaces. Application areas include weighted estimates for the heat kernel, shape optimization and properties of very weak solutions to NavierStokes problems in weighted spaces and various type of domains.



The motion of rigid bodies in liquid: mathematical analysis, numerical simulation and related problems (IAA100190804)
from 01/01/2008
to 31/12/2010 main investigator

In the framework of the project we will study the steady flow around bodies. We will consider the case when the direction of the angular velocity and of the velocity at infinity are or are not parallel. We will extend the results from the previous project, where the angular and tranlation velocities were parallel. We will study the linear cases and NavierStokes equations. We will investigate the existence of solution, asymptotic behaviour, resolvent and spectrum problem. Further, we will study the motion of several bodies in the fluid. We will consider the influence of boundary conditions and possibility of collisions. In this part we will study the existence of weak solution for steady and nonsteady cases. We will investigate fluid flows described by NavierStokes equations as well as by nonNewtonian models. We will investigate the modeling of blood flow and related cardiovascular cases. Next to it the numerical simulation of severeal models will be performed.



Qualitative analysis and numerical solution of flow problems (201/08/0012)
from 01/01/2008
to 31/12/2012 investigator

Objectives:
Mathematical modelling of fluid flows in different regimes.



Nečas Center for Mathematical Modeling  part IM (LC06052)
from 01/01/2006
to 31/12/2011 investigator

The general goal of the Nečas Center for Mathematical Modeling is to establish a significant scientific team in the field of mathematical properties of models in continuum mechanics and thermodynamics, developed by an intensive collaboration of five important research teams at three Prague affiliations and their goaldirected collaboration with top experts from abroad. The research projects of the center include: 1) Nonlinear theoretical, numerical and computer analysis of problems of continuum physics. 2) Heatconductive and deforming processes in compressible fluids, incompressible substances of fluid type, and in linearly elastic matters. 3) Interaction of the substances. 4) Biochemical procedures in substances. 5) Passages between models, dimensional analysis.



Mathematical modelling of motion of bodies in Newtonian and nonNewtonian fluids and related mathematical problems (IAA100190505)
from 01/01/2005
to 01/12/2007 main investigator

Investigation of properties of models describing motion of rigid bodies in viscous fluid. Existence of weak and strong solutions, asymptotic behaviour, attainability, numerical analysis and solution of selected models.



Mathematical theory and numerical simulation of problems in the fluid mechanics (201/05/0005)
from 01/01/2005
to 31/12/2007 investigator

The goal of this project is to investigate various models of the fluid mechanics from the theoretiacl point of view (exitence, uniqueness, regularity) and to use the theoretical results to improving various numerical methods in the fluid flow modeling.



Mathematical and numerical analysis of problems in fluid mechanics (201/02/0684)
from 01/01/2002
to 31/12/2004 investigator

The project includes several topics: 1) Regularity of weak solutions to the NavierStokes equations. 2) Existence, stability and long time behaviour of solutions to compressible NavierStokes equations. 3) Interaction of fluids with rigid bodies. 4) Qualitative theory of higherdegree fluids. 5) Higher order schemes for compressible flows. 6) Adaptive methods of mesh refinement near discontinuities. 7) Extensions of finite volume  finite element methods from 2D to 3D problems.
