
Solving ill posed problems in the dynamics of compressible fluids (2102411S)
from 01/01/2021
to 31/12/2023 main investigator

Objectives:
The project focuses on mathematical models of compressible fluids that are illposed in the framework of the existing theory. A well known example is the Euler system describing a compressible perfect gas. By solving them we mean developing suitable consistent approximation, identifying the class of limits of approximate solutions, and designing appropriate numerical methods.



Mathematics of diffuse interface models (8J20FR007)
from 01/01/2020
to 31/12/2021 main investigator

The aim of the project is qualitative study of complex compressible fluid models with difuse interfaces, subject to stochastic external forces. In particular, the following will be addressed:
Existence of a suitable class of weak solutions
Relative (weakstrong) uniqueness
Singular limits
Desing, implementation and analysis of numerical schemes



Oscillations and concentrations versus stability in the equations of mathematical fluid dynamics (GA1805974S)
from 01/01/2018
to 30/06/2021 main investigator

Objectives:
The project focuses on questions of possible singularities in the equations of mathematical fluid dynamics and their adequate description by means of weak and measure valued solutions. The main topics include:(i) dissipative solutions, (ii) admissibility criteria, (iii) equations with stochastic terms, (iv) applications in the numerical analysis.
The goal is to develop a consistent mathematical theory of fluids in motion in the framework of weak and measure valued solutions, developing the concept of dissipative solution, obtaining new admissibility criteria, solving problems with stochastic terms, analyzing the underlying numerical schemes.



Dynamics of mutlicomponent fluids (7AMB17FR053)
from 01/01/2017
to 31/12/2018 main investigator

The goal of the project is studying qualitative properties of a particular class of the socalled energetically weak solutions to complex system of the NavierStokesFourier type as well as Coupling of these systems with the phase transition equations of the CahnHilliard or AllenCahn type. We plan to investigate these problems also in unbounded physical domains in appropriate classes of uniformly bounded functions.
The main goal is obtaining new results in the following directions:
• applications of the relative entropy methods and the consequences concerning stability of the socalled dissipative solutions
• singular limits, in particular the sharp interface limits with rigorous mathematical justification
• longtime dynamics, with a particular emphasis on the existence of bounded absorbing sets, asymptotic compactness of greajectories and the relevant questions concerning the attractors and their structure and complexity



Mathematical Thermodynamics of Fluids (MATHEF(320078))
from 01/05/2013
to 30/04/2018 main investigator

Programme type: FP7 ERC Advanced Grant
Objectives:
The main goal of the present research proposal is to build up a general mathematical theory describing the motion of a compressible, viscous, and heat conductive fluid. Our approach is based on the concept of generalized (weak) solutions satisfying the basic physical principles of balance of mass, momentum, and energy. The energy balance is expressed in terms of a variant of entropy inequality supplemented with an integral identity for the total energy balance.
We propose to identify a class of suitable weak solutions, where admissibility is based on a direct application of the principle of maximal entropy production compatible with Second law of thermodynamics. Stability of the solution family will be investigated by the method of relative entropies constructed on the basis of certain thermodynamic potentials as ballistic free energy.
The new solution framework will be applied to multiscale problems, where several characteristic scales become small or extremely large. We focus on mutual interaction of scales during this process and identify the asymptotic behavior of the quantities that are filtered out in the singular limits. We also propose to study the influence of the geometry of the underlying physical space that may change in the course of the limit process. In particular, problems arising in homogenization and optimal shape design in combination with various singular limits are taken into account.
The abstract approximate scheme used in the existence theory will be adapted in order to develop adequate numerical methods. We study stability and convergence of these methods using the tools developed in the abstract part, in particular, the relative entropies.



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 main 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.



Mathematical and computer analysis of the evolution processes in nonlinear viscoelastic fluidlike materials (201/09/0917)
from 01/01/2009
to 31/12/2013 investigator

Objectives:
This project proposal focuses on theoretical and computer analysis, and their mutual interplay, related to several classes of evolutionary models that have been recently designed to capture complex behavior of various fluidlike materials within the framework of nonlinear continuum mechanics. The characteristic keywords of these particular classes are implicit constitutive relations, nonlinear rate type fluids, nonlinear integral type fluidlike materials, inhomogeneous incompressible fluids, compressible nonNewtonian fluids, and chemically reacting fluids. Regarding specific applications, we intent to concentrate on unsteady flows of biological liquids and timedependent processes in geophysical materials. The goal is to develop new methods and tools to solve initial boundaryvalue problems for large data, both theoretically and numerically.



Mathematical analysis of complex systems in the fluid mechanics (201/08/0315)
from 01/01/2008
to 31/12/2011 main investigator

The main goal of the project is to develop a rigorous mathematical theory of complex systems in fluid mechanics. Such problems arise in models of chemical reactions, astrophysics, biological models, atmosphere and geophysical fluid dynamics. The main challenge here is to handle problems with large data and without any restriction concerning the time scale. The main topics include: Multicomponent problems and mixtures. 2. Equations of magnetohydrodynamics. 3. Atmospheric and geophysical models. 4. Large time behavior of solutions and equilibrium states.



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



Asymptotic analysis of infinite dimensional dynamical systems (IAA100190606)
from 01/01/2006
to 31/12/2008 main investigator

The goal of the project is to obtain new qualitative results concerning the asymptotic behavior of infinite dimensional dynamical systems arising especially in the theory of viscous compressible fluids. The main topics include compactness of solutions, global existence, convergence towards equilibria and problems with rapidly oscillating boundaries.



Mathematical analysis in the thermodynamics of fluids (201/05/0164)
from 01/01/2005
to 30/12/2007 main investigator

The aim of the present research project is to establish a coherent mathematical theory of viscous heat conducting fluids based on a suitable variational formulation of the problem consistent with the second law of thermodynamics. The main topics include: 1. The existence of solutions on arbitrarily large time intervals with no restriction on the size of data. 2. The questions of uniqueness, boundedness, and stability of solutions with respect to the initial conditions and other parameters as the case may be. 3. The long time behavior, convergence towards equilibria, and attractors. 4. Sensitivity analysis with respect to the shape of the underlying spatial domain.



Compatibility of dynamics and statics in multicomponent dissipative systems (IAA1019302)
from 01/01/2003
to 01/01/2005 main investigator

The main topics of the project is to study the asymptotic behaviour of solutions to partial differential equations arising in multicomponent systems modelling. The long time behaviour of solutions as well as the problem of stabilization towards stationary state will be investigated. Specifically, we shall investigate: 1. The equations describing the motion of one or several rigid bodies in a viscous fluid. 2. The solidliquid phase fields models. 3. Dynamical solidsolid phase transition models.
