Eduard Feireisl

On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations

Abstract. We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies

On the motion of rigid bodies in a viscous compressible fluid

\gamma > 3/2

On the Navier-Stokes equations with temperature-dependent transport coefficients

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ANALYSIS OF A PHASE-FIELD MODEL FOR TWO-PHASE COMPRESSIBLE FLUIDS

We prove the existence of global-in-time weak solutions to a model describing the motion of several rigid bodies in a viscous compressible fluid. Unlike most recent results of similar type, there is no restriction on the existence time, regardless of possible collisions of two or more rigid bodies and/or a contact of the bodies with the boundary.

A new approach to non-isothermal models for nematic liquid crystals

We establish long-time and large-data existence of a weak solution to the problem describing three-dimensional unsteady flows of an incompressible fluid, where the viscosity and heat-conductivity coefficients vary with the temperature. The approach reposes on considering the equation for the total energy rather than the equation for the temperature. We consider the spatially periodic problem.

Long-time stabilization of solutions to a phase-field model with memory

A model describing the evolution of a binary mixture of compressible, viscous, and macroscopically immiscible fluids is investigated. The existence of global-in-time weak solutions for the resulting system coupling the compressible Navier–Stokes equations governing the motion of the mixture with the Allen–Cahn equation for the order parameter is proved without any restriction on the size of initial data.

Inviscid Incompressible Limits of the Full Navier-Stokes-Fourier System

We introduce a new class of non-isothermal models describing the evolution of nematic liquid crystals and prove their consistency with the fundamental laws of classical thermodynamics. The resulting system of equations captures all essential features of physically relevant models; in particular, the effect of stretching of the director field is taken into account. In addition, the associated initial-boundary value problem admits global-in-time weak solutions without any essential restrictions on the size of the initial data.

On a non-isothermal model for nematic liquid crystals

Abstract. We prove that any global bounded solution of a phase field model with memory terms tends to a single equilibrium state for large times. Because of the memory effects, the energy is not a Lyapunov function for the problem and the set of equilibria may contain a nontrivial continuum of stationary states. The method we develop is applicable to a more general class of equations containing memory terms.

Stability of Flows of Real Monoatomic Gases

We consider the full Navier-Stokes-Fourier system in the singular limit for the small Mach and large Reynolds and Péclet numbers, with ill prepared initial data on R3. The Euler-Boussinesq approximation is identified as the limit system.

On the weak solutions to the equations of a compressible heat conducting gas

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the absolute temperature , the velocity field u and the director field d, representing preferred orientation of molecules in a neighbourhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier–Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field d, where the transport (viscosity) coefficients vary with temperature. The dynamics of d is described by means of a parabolic equation of Ginzburg–Landau type, with a suitable penalization term to relax the constraint |d| = 1. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fouriers law, depending also on the director field d. The proposed model is shown to be compatible with first and second laws of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data.