
Compressible NavierStokesFourier system for the adiabatic coefficient close to one
Skříšovský, Emil ; Pokorný, Milan (advisor) ; Feireisl, Eduard (referee)
In the present thesis we study the compressible NavierStokesFourier sys tem. This is a system of partial differential equations describing the evolutionary problem for an adiabatic flow of a heat conducting compressible viscous fluid in a bounded domain. Here we consider the problem in two dimensions with zero Dirichlet boundary conditions for velocity. The cold pressure term in the pressure law for the momentum equation is here considered in the form pC(ϱ) ∼ ϱ logα (1+ϱ) for some α > 0, for which we need to work on the scale of Orlicz spaces in order to obtain useful estimates and in those space we formulate the problem weakly and also establish the weak compactness of the solution. The main result of this thesis is Theorem 6.1 where we show the existence of a weak solution with no assumptions on the size of the data and on arbitrary large time intervals. 1


Mathematical Thermodynamics of Viscous Fluids
Feireisl, Eduard
This course is a short introduction to the mathematical theory of the motion of viscous fluids. We introduce the concept of weak solution to the NavierStokesFourier system and discuss its basic properties. In particular, we construct the weak solutions as a suitable limit of a mixed numerical scheme based on a combination of the finite volume and finite elements method. The question of stability and robustness of various classes of solutions is addressed with the help of the relative (modulated) energy functional. Related results concerning weakstrong uniqueness and conditional regularity of weak solutions are presented. Finally, we discuss the asymptotic limit when viscosity of the fluid tends to zero. Several examples of ill posedness for the limit Euler system are given and an admissibility criterion based on the viscous approximation is proposed.

 

Mathematical analysis of fluids in motion
Michálek, Martin ; Feireisl, Eduard (advisor) ; Wiedemann, Emil (referee) ; Swierczewska  Gwiazda, Agnieszka (referee)
The aim of this work is to provide new results of global existence for dif ferent evolution equations of fluid mechanics. We are in general interested in finding weak solutions without restrictions on the size of initial data. The proofs of existence are based on several different approaches including en ergy methods, convergence analysis of finite numerical methods and convex integration. All these techniques significantly exploit results of mathematical analysis and other branches of mathematics. 1


Mathematical analysis of equations describing the flow of compressible heat conducting fluids
Axmann, Šimon ; Pokorný, Milan (advisor) ; Feireisl, Eduard (referee) ; Novotný, Antonín (referee)
Title: Mathematical analysis of equations describing the flow of compressible heat conducting fluids Author: Šimon Axmann Department: Mathematical Institute of Charles University Supervisor: doc. Mgr. Milan Pokorný, Ph.D., Mathematical Institute of Charles University Abstract: The present thesis is devoted to the mathematical analysis of equa tions describing the flow of viscous compressible newtonian fluid in various time regimes. In particular, we present existence results for three problems arising as special cases of a general model derived in the introductory part. The first chap ter deals with timeperiodic solutions to the full NavierStokesFourier system for heatconducting fluid. The second chapter contains the proof of existence of steady solutions to a system arising from phase field model for twophase com pressible fluid. Finally, in the last section we study steady strong solutions to the NavierStokes equations under the additional assumption that the fluid is suffi ciently dense. For each problem a different concept of the solution is considered, on the other hand in all cases an essential role is played by the crucial quantity effective viscous flux. Keywords: compressible NavierStokes system; weak solution; entropy variational solution; large data


Analysis of dissipative equations in unbounded domains
Michálek, Martin ; Pražák, Dalibor (advisor) ; Feireisl, Eduard (referee)
In the first part of this thesis, suitable function spaces for analysis of partial differ ential equations in unbounded domains are introduced and studied. The results are then applied in the second part on semilinear wave equation in Rd with non linear source term and nonlinear damping. The source term is supposed to be bounded by a polynomial function with a subcritical growth. The damping term is strictly monotone and satisfying a polynomiallike growth condition. Global existence is proved using finite speed of propagation. Dissipativity in locally uni form spaces and the existence of a locally compact attractor are then obtained after additional conditions imposed on the damping term.


Compressible fluid motion in time dependent domains
Sýkora, Petr ; Feireisl, Eduard (advisor) ; Pokorný, Milan (referee)
In this work we study the existence of weak solutions for compressible NavierStokes equations in unbounded time dependent domains. Using the methods introduced in Feireisl E. Dynamics of Viscous Compressible Fluids we extend the results of article Feireisl E. Neustupa J. Stebel J., Convergence of a Brinkmantype penalization for compressible fluid flows, which studies the flow with a "noslip" boundary condition on bounded domains. Next, we extend results of article Feireisl E. Kreml O. Nečasová Š. Neustupa J. Stebel J., Weak solutions to the barotropic Navier Stokes system with slip boundary conditions in time dependent domains, which studies flow with compete Navier boundary condition. Finally, we discuss solutions for rotating fluid system. In this case, there are new members in momentum equation, representing the Coriolis and centrifugal force, which cause problems.


Mathematical Analysis of Fluids in Large Domains
Poul, Lukáš ; Feireisl, Eduard (advisor) ; Pokorný, Milan (referee) ; Vodák, Rostislav (referee)
This thesis contains a set of articles concerned with flow of a viscous, compressible and heat conducting fluid in large domains. In the first part of the thesis, the existence of the weak solutions in unbounded domains is studied. The results follow each other in the way they were obtained through the time, and range from a simple extension to bounded domains with Lipschitz boundary up to the most general existence theorem for fluid flow in general open sets. The existence results are supplemented with the study of existence of weak solutions in the unbounded domain case with prescribed nonvanishing boundary conditions for density and temperature at infinity. The last contribution then concerns with the low Mach number limit in the compressible fluid flow.

 
 