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



Mathematical analysis of partial differential equations describing inviscid flows (GJ1701694Y)
from 01/01/2017
to 31/12/2019 main investigator

Objectives:
The investigator and his team will develop the theory of Camillo De Lellis and Lászlo Székelyhidi which allows to prove surprising results for incompressible Euler equations in multiple space dimensions. They will focus mainly on development and applications of the theory in the field of compressible flow, both in the isentropic case and in the case of full system of partial differential equations. They will study and propose criteria to choose "physical" solutions among the infinitely many weak solutions of appropriate systems of equations. The investigator and his team will maintain already established scientific cooperations (De Lellis, Chiodaroli, Wiedemann) and establish new ones. The results of the project will be presented on international conferences and will be published as articles in impacted journals.



Mathematical analysis of hyperbolic conservation laws (Neuron Impuls Junior)
from 01/01/2017
to 30/06/2018 main investigator

Objectives:
The goal of the project is to deepen present knowledge concerning existence of weak solutions for hyperbolic systems of partial differential equations arising from conservation laws in multiple space dimensions, study of uniqueness and nonuniqueness of entropic weak solutions, analysis of admissibility criteria (maximization of entropy production or inviscid limit) or even designing other suitable admissibility criteria.
A convenient test problem for admissibility criteria is the so called Riemann problem, i.e. problem with initial data consisting of a jump discontinuity separating two constant states. As such initial data are onedimensional, one can use standard theory to find a onedimensional selfsimilar solution which is unique in the class of BV functions. In a broader class of functions (in particular in the class of essentialy bounded functions), other weak solutions may exist as is in the case of compressible Euler equations. In the framework of this project I will study also Riemann problems for various hyperbolic systems concerning uniqueness or nonuniqueness of weak solutions in the class of essentialy bounded functions and in the case of nonuniqueness, admissibility criteria will be tested.



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



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



Transport phenomena in continuum fluid dynamics (TraFlu(SCIEX 11.152))
from 01/07/2012
to 31/12/2013 main investigator

Sciex Postdoctoral Fellowship for Ondřej Kreml at University of Zurich (Host institution).
Objectives:
Ondřej Kreml will study the results of Camillo De Lellis and László Székelyhidi about illposedness of bounded weak solutions for the incompressible Euler equations and bounded admissible solutions for the compressible isentropic Euler system in multiple space dimensions. The objectives of the project are to generalize the illposedness results for compressible isentropic Euler system and to study the Riemann problem for this system. Another objective is to modify the method of De Lellis and Székelyhidi to be applicable in other systems of partial differential equations describing inviscid fluid flows.



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 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.



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.
