DCDS-S
A stabilizing effect of a high-frequency driving force on the motion of a viscous, compressible, and heat conducting fluid
Eduard Feireisl Dalibor Pražák
Discrete & Continuous Dynamical Systems - S 2009, 2(1): 95-111 doi: 10.3934/dcdss.2009.2.95
We study the impact of an oscillating external force on the motion of a viscous, compressible, and heat conducting fluid. Assuming that the frequency of oscillations increases sufficiently fast as the time goes to infinity, the solutions are shown to stabilize to a spatially homogeneous static state.
keywords: heat conducting fluid high-frequency oscillations Compressible fluid stabilization.
DCDS-S
On weak solutions to a diffuse interface model of a binary mixture of compressible fluids
Eduard Feireisl
Discrete & Continuous Dynamical Systems - S 2016, 9(1): 173-183 doi: 10.3934/dcdss.2016.9.173
We consider the Euler-Cahn-Hilliard system proposed by Lowengrub and Truskinovsky describing the motion of a binary mixture of compressible fluids. We show that the associated initial-value problem possesses infinitely many global-in-time weak solutions for any finite energy initial data. A modification of the method of convex integration is used to prove the result.
keywords: weak solution diffuse interface model. Euler-Cahn-Hilliard system
DCDS-S
On the motion of incompressible inhomogeneous Euler-Korteweg fluids
Miroslav Bulíček Eduard Feireisl Josef Málek Roman Shvydkoy
Discrete & Continuous Dynamical Systems - S 2010, 3(3): 497-515 doi: 10.3934/dcdss.2010.3.497
We study a system of equations governing evolution of incompressible inhomogeneous Euler-Korteweg fluids that describe a class of incompressible elastic materials. A local well-posedness theory is developed on a bounded smooth domain with no-slip boundary condition on velocity and vanishing gradient of density. The cases of open space and periodic box are also considered, where the local existence and uniqueness of solutions is shown in Sobolev spaces up to the critical smoothness $\frac{n}{2}+1$.
keywords: local-in-time well-posedness smooth solution. Korteweg stress Korteweg fluid inhomogeneous Euler fluid
DCDS
Low Mach number asymptotics for reacting compressible fluid flows
Eduard Feireisl Hana Petzeltová
Discrete & Continuous Dynamical Systems - A 2010, 26(2): 455-480 doi: 10.3934/dcds.2010.26.455
We study the low Mach number limit for the full Navier-Stokes-Fourier system describing the dynamics of chemically reacting fluids. The so-called reactive Boussinesq system is identified as the asymptotic limit.
keywords: Low Mach number Navier-Stokes-Fourier system reacting fluids.
DCDS
Global in time weak solutions for compressible barotropic self-gravitating fluids
Bernard Ducomet Eduard Feireisl Hana Petzeltová Ivan Straškraba
Discrete & Continuous Dynamical Systems - A 2004, 11(1): 113-130 doi: 10.3934/dcds.2004.11.113
The existence of global in time weak solutions to the Navier-Stokes-Poisson system of barotropic compressible flow is proved. The system takes into account the effect of self-gravitation. Moreover, the case of a non-monotone pressure important in certain applications in astrophysics and the theory of nuclear fluids is included.
keywords: global in time solutions. Compressible self-gravitating fluid Navier-Stokes-Poisson system
DCDS-S
New developments in mathematical theory of fluid mechanics
Eduard Feireisl Šárka Nečasová Reimund Rautmann Werner Varnhorn
Discrete & Continuous Dynamical Systems - S 2014, 7(5): i-ii doi: 10.3934/dcdss.2014.7.5i
Mathematical theory of fluid mechanics is a field with a rich long history and active present. The volume collects selected contributions of distinguished experts in various domains ranging from modeling through mathematical analysis to numerics and practical implementations related to real world problems.

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DCDS
Long time behavior and attractors for energetically insulated fluid systems
Eduard Feireisl
Discrete & Continuous Dynamical Systems - A 2010, 27(4): 1587-1609 doi: 10.3934/dcds.2010.27.1587
We study the long time behavior, and, in particular, the existence of attractors for the Navier-Stokes-Fourier system under energetically insulated boundary conditions. We show that the attractor consists of static solutions determined uniquely by the total mass and energy of the fluid.
keywords: attractor long time behavior. Navier-Stokes-Fourier system
DCDS
Relative entropies in thermodynamics of complete fluid systems
Eduard Feireisl
Discrete & Continuous Dynamical Systems - A 2012, 32(9): 3059-3080 doi: 10.3934/dcds.2012.32.3059
We introduce the notion of relative entropy in the framework of thermodynamics of compressible, viscous and heat conducting fluids. The relative entropy is constructed on the basis of a thermodynamic potential called ballistic free energy and provides stability of solutions to the associated Navier-Stokes-Fourier system with respect to perturbations. The theory is illustrated by applications to problems related to the long time behavior of solutions and the problem of weak-strong uniqueness.
keywords: relative entropy qualitative properties of compressible fluids. Navier-Stokes-Fourier system
DCDS
Mathematical theory of viscous fluids: Retrospective and future perspectives
Eduard Feireisl
Discrete & Continuous Dynamical Systems - A 2010, 27(2): 533-555 doi: 10.3934/dcds.2010.27.533
We review the recent state of art of the mathematical theory of viscous, compressible, and heat conducting fluids. We emphasize the significant role of the Second law of thermodynamics in our approach. Qualitative properties of solutions and relations between different models are also discussed.
keywords: qualitative properties of compressible fluids. global existence Navier-Stokes-Fourier system
DCDS-S
Preface
Eduard Feireisl Mirko Rokyta Josef Málek
Discrete & Continuous Dynamical Systems - S 2008, 1(3): i-iii doi: 10.3934/dcdss.2008.1.3i
The Tenth International School on Mathematical Theory in Fluid Mechanics was held at the small village of Paseky in the northern part of the Czech Republic in May 11--18, 2007. The main part of the program of the school consisted of series of lectures delivered by Thomas Alazard, Camillo De Lellis, Eduard Feireisl, Isabelle Gallagher, and Herbert Koch. The presented book contains five survey contributions, based on the respective series of lectures.
       The article "A minicourse on the low Mach number limit" by Thomas Alazard is devoted to the study of the low Mach number limit for classical solutions of the compressible Navier-Stokes or Euler equations for non-isentropic flows. The general case is studied, in which the combined effects of large temperature variations and thermal conduction are taken into account.

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