Miroslav Krbec

Grants
- Flow of fluids in domains with variable geometry (P201/11/1304)
  from 01/01/2011 to 31/12/2013 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 non-Newtonian 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 Navier-Stokes systems. Specifically we intend to study imbeddings, traces ad interpolation inequalities in general spaces of Besov and Lizorkin-Triebel 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 Navier-Stokes 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 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 Navier-Stokes 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 non-steady cases. We will investigate fluid flows described by Navier-Stokes equations as well as by non-Newtonian 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.
- Modern methods in function spaces and applications (201/06/0400)
  from 01/01/2006 to 31/12/2008 main investigator
- The project continues the research of the applicant in last years and it is focused on topical problems in function spaces relevant for applications in differential equations. The major goals are: extrapolation properties of Lebesgue and Orlicz spaces, including their weighted clones, differentiability properties of operators in them, associated extrapolation behaviour of traces of functions in Sobolev spaces, imbedding properties of anisotropic spaces of Sobolev type via real and Fourier analysis techniques with emphasis on spaces with dominating mixed smoothness.
- 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 goal-directed 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) Heat-conductive 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 non-Newtonian fluids and related mathematical problems (IAA100190505)
  from 01/01/2005 to 01/12/2007 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.