Šárka Nečasová

Grants
- Thermodynamically consistent models for fluid flows: mathematical theory and numerical solution (GA16-03230S)
  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 well-posedness 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.
- Flow of viscous fluid in time dependent domain (7AMB16PL060)
  from 01/01/2016 to 31/12/2017 main investigator
- Objectives: G1. Global existence of weak solution of full system in time-dependent 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 time-dependent domain. We will use the results from fixed domain introduced in E. Feireisl and A. Novotný: Weak-strong uniqueness for the full Navier-Stokes-Fourier 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 time-dependent 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.
- Qualitative analysis and numerical solution of problems of flows in generally time-dependent domains with various boundary conditions (GA13-00522S)
  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 Navier-Stokes equations in fixed or time-dependent 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 hp-methods, will be tested on suitable problems and applied to fluid-structure 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 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 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 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.
- 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 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 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 Navier-Stokes equations. 2) Existence, stability and long time behaviour of solutions to compressible Navier-Stokes equations. 3) Interaction of fluids with rigid bodies. 4) Qualitative theory of higher-degree 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.