On weak solutions to a diffuse interface model of a binary mixture of compressible fluids
Eduard Feireisl
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
On the motion of incompressible inhomogeneous Euler-Korteweg fluids
Miroslav Bulíček Eduard Feireisl Josef Málek Roman Shvydkoy
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
Low Mach number asymptotics for reacting compressible fluid flows
Eduard Feireisl Hana Petzeltová
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.
Global in time weak solutions for compressible barotropic self-gravitating fluids
Bernard Ducomet Eduard Feireisl Hana Petzeltová Ivan Straškraba
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
New developments in mathematical theory of fluid mechanics
Eduard Feireisl Šárka Nečasová Reimund Rautmann Werner Varnhorn
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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Long time behavior and attractors for energetically insulated fluid systems
Eduard Feireisl
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
Relative entropies in thermodynamics of complete fluid systems
Eduard Feireisl
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
Mathematical theory of viscous fluids: Retrospective and future perspectives
Eduard Feireisl
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
A stabilizing effect of a high-frequency driving force on the motion of a viscous, compressible, and heat conducting fluid
Eduard Feireisl Dalibor Pražák
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.
Polynomial stabilization of some dissipative hyperbolic systems
Kais Ammari Eduard Feireisl Serge Nicaise
We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
keywords: observability inequality polynomial stability dissipative hyberbolic system acoustic equation. Exponential stability resolvent estimate

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