525 research outputs found
Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity
We consider a Navier-Stokes-Voigt fluid model where the instantaneous
kinematic viscosity has been completely replaced by a memory term incorporating
hereditary effects, in presence of Ekman damping. The dissipative character of
our model is weaker than the one where hereditary and instantaneous viscosity
coexist, previously studied by Gal and Tachim-Medjo. Nevertheless, we prove the
existence of a regular exponential attractor of finite fractal dimension under
rather sharp assumptions on the memory kernel.Comment: 26 page
Analyticity and Gevrey-class regularity for the second-grade fluid equations
We address the global persistence of analyticity and Gevrey-class regularity
of solutions to the two and three-dimensional visco-elastic second-grade fluid
equations. We obtain an explicit novel lower bound on the radius of analyticity
of the solutions to the second-grade fluid equations that does not vanish as
. Applications to the damped Euler equations are given
On a Navier-Stokes-Allen-Cahn model with inertial effects
A mathematical model describing the flow of two-phase fluids in a bounded
container is considered under the assumption that the phase transition
process is influenced by inertial effects. The model couples a variant of the
Navier-Stokes system for the velocity with an Allen-Cahn-type equation for
the order parameter relaxed in time in order to introduce inertia.
The resulting model is characterized by second-order material derivatives which
constitute the main difficulty in the mathematical analysis. Actually, in order
to obtain a tractable problem, a viscous relaxation term is included in the
phase equation. The mathematical results consist in existence of weak solutions
in 3D and, under additional assumptions, existence and uniqueness of strong
solutions in 2D. A partial characterization of the long-time behavior of
solutions is also given and in particular some issues related to dissipation of
energy are discussed.Comment: 24 page
Compressible fluids interacting with a linear-elastic shell
We study the Navier--Stokes equations governing the motion of an isentropic
compressible fluid in three dimensions interacting with a flexible shell of
Koiter type. The latter one constitutes a moving part of the boundary of the
physical domain. Its deformation is modeled by a linearized version of Koiter's
elastic energy. We show the existence of weak solutions to the corresponding
system of PDEs provided the adiabatic exponent satisfies
( in two dimensions). The solution exists until the moving boundary
approaches a self-intersection. This provides a compressible counterpart of the
results in [D. Lengeler, M. \Ruzicka, Weak Solutions for an Incompressible
Newtonian Fluid Interacting with a Koiter Type Shell. Arch. Ration. Mech. Anal.
211 (2014), no. 1, 205--255] on incompressible Navier--Stokes equations
The Cosmic No-Hair Theorem and the Nonlinear Stability of Homogeneous Newtonian Cosmological Models
The validity of the cosmic no-hair theorem is investigated in the context of
Newtonian cosmology with a perfect fluid matter model and a positive
cosmological constant. It is shown that if the initial data for an expanding
cosmological model of this type is subjected to a small perturbation then the
corresponding solution exists globally in the future and the perturbation
decays in a way which can be described precisely. It is emphasized that no
linearization of the equations or special symmetry assumptions are needed. The
result can also be interpreted as a proof of the nonlinear stability of the
homogeneous models. In order to prove the theorem we write the general solution
as the sum of a homogeneous background and a perturbation. As a by-product of
the analysis it is found that there is an invariant sense in which an
inhomogeneous model can be regarded as a perturbation of a unique homogeneous
model. A method is given for associating uniquely to each Newtonian
cosmological model with compact spatial sections a spatially homogeneous model
which incorporates its large-scale dynamics. This procedure appears very
natural in the Newton-Cartan theory which we take as the starting point for
Newtonian cosmology.Comment: 16 pages, MPA-AR-94-
Stabilizing the Long-time Behavior of the Navier-Stokes Equations and Damped Euler Systems by Fast Oscillating Forces
The paper studies the issue of stability of solutions to the Navier-Stokes
and damped Euler systems in periodic boxes. We show that under action of fast
oscillating-in- time external forces all two dimensional regular solutions
converge to a time periodic flow. Unexpectedly, effects of stabilization can be
also obtained for systems with stationary forces with large total momentum
(average of the velocity). Thanks to the Galilean transformation and space
boundary conditions, the stationary force changes into one with time
oscillations. In the three dimensional case we show an analogical result for
weak solutions to the Navier- Stokes equations
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