## Journals

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DCDS-S

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.

DCDS-S

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 $\frac{n}{2}+1$.

DCDS

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.

DCDS

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.

DCDS-S

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.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

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DCDS

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.

DCDS

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.
DCDS

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.

DCDS-S

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.

DCDS

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.

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