697 research outputs found
Incompressible flow in porous media with fractional diffusion
In this paper we study the heat transfer with a general fractional diffusion
term of an incompressible fluid in a porous medium governed by Darcy's law. We
show formation of singularities with infinite energy and for finite energy we
obtain existence and uniqueness results of strong solutions for the
sub-critical and critical cases. We prove global existence of weak solutions
for different cases. Moreover, we obtain the decay of the solution in ,
for any , and the asymptotic behavior is shown. Finally, we prove the
existence of an attractor in a weak sense and, for the sub-critical dissipative
case with , we obtain the existence of the global attractor
for the solutions in the space for any
Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain
In this paper we deal with the local null controllability of the
N-dimensional Navier-Stokes system with internal controls having one vanishing
component. The novelty of this work is that no condition is imposed on the
control domain
Breakdown of Conformal Invariance at Strongly Random Critical Points
We consider the breakdown of conformal and scale invariance in random systems
with strongly random critical points. Extending previous results on
one-dimensional systems, we provide an example of a three-dimensional system
which has a strongly random critical point. The average correlation functions
of this system demonstrate a breakdown of conformal invariance, while the
typical correlation functions demonstrate a breakdown of scale invariance. The
breakdown of conformal invariance is due to the vanishing of the correlation
functions at the infinite disorder fixed point, causing the critical
correlation functions to be controlled by a dangerously irrelevant operator
describing the approach to the fixed point. We relate the computation of
average correlation functions to a problem of persistence in the RG flow.Comment: 9 page
On a non-isothermal model for nematic liquid crystals
A model describing the evolution of a liquid crystal substance in the nematic
phase is investigated in terms of three basic state variables: the {\it
absolute temperature} \teta, the {\it velocity field} \ub, and the {\it
director field} \bd, representing preferred orientation of molecules in a
neighborhood 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 \bd, where the transport (viscosity) coefficients vary
with temperature. The dynamics of \bd is described by means of a parabolic
equation of Ginzburg-Landau type, with a suitable penalization term to relax
the constraint |\bd | = 1. The system is supplemented by a heat equation,
where the heat flux is given by a variant of Fourier's law, depending also on
the director field \bd. The proposed model is shown compatible with
\emph{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
Geometric shape of invariant manifolds for a class of stochastic partial differential equations
Invariant manifolds play an important role in the study of the qualitative
dynamical behaviors for nonlinear stochastic partial differential equations.
However, the geometric shape of these manifolds is largely unclear. The purpose
of the present paper is to try to describe the geometric shape of invariant
manifolds for a class of stochastic partial differential equations with
multiplicative white noises. The local geometric shape of invariant manifolds
is approximated, which holds with significant likelihood. Furthermore, the
result is compared with that for the corresponding deterministic partial
differential equations
Chaos in computer performance
Modern computer microprocessors are composed of hundreds of millions of
transistors that interact through intricate protocols. Their performance during
program execution may be highly variable and present aperiodic oscillations. In
this paper, we apply current nonlinear time series analysis techniques to the
performances of modern microprocessors during the execution of prototypical
programs. Our results present pieces of evidence strongly supporting that the
high variability of the performance dynamics during the execution of several
programs display low-dimensional deterministic chaos, with sensitivity to
initial conditions comparable to textbook models. Taken together, these results
show that the instantaneous performances of modern microprocessors constitute a
complex (or at least complicated) system and would benefit from analysis with
modern tools of nonlinear and complexity science
Optimal Constraint Projection for Hyperbolic Evolution Systems
Techniques are developed for projecting the solutions of symmetric hyperbolic
evolution systems onto the constraint submanifold (the constraint-satisfying
subset of the dynamical field space). These optimal projections map a field
configuration to the ``nearest'' configuration in the constraint submanifold,
where distances between configurations are measured with the natural metric on
the space of dynamical fields. The construction and use of these projections is
illustrated for a new representation of the scalar field equation that exhibits
both bulk and boundary generated constraint violations. Numerical simulations
on a black-hole background show that bulk constraint violations cannot be
controlled by constraint-preserving boundary conditions alone, but are
effectively controlled by constraint projection. Simulations also show that
constraint violations entering through boundaries cannot be controlled by
constraint projection alone, but are controlled by constraint-preserving
boundary conditions. Numerical solutions to the pathological scalar field
system are shown to converge to solutions of a standard representation of the
scalar field equation when constraint projection and constraint-preserving
boundary conditions are used together.Comment: final version with minor changes; 16 pages, 14 figure
Existence and equilibration of global weak solutions to Hookean-type bead-spring chain models for dilute polymers
We show the existence of global-in-time weak solutions to a general class of
coupled Hookean-type bead-spring chain models that arise from the kinetic
theory of dilute solutions of polymeric liquids with noninteracting polymer
chains. The class of models involves the unsteady incompressible Navier-Stokes
equations in a bounded domain in two or three space dimensions for the velocity
and the pressure of the fluid, with an elastic extra-stress tensor appearing on
the right-hand side in the momentum equation. The extra-stress tensor stems
from the random movement of the polymer chains and is defined by the Kramers
expression through the associated probability density function that satisfies a
Fokker-Planck-type parabolic equation, a crucial feature of which is the
presence of a center-of-mass diffusion term. We require no structural
assumptions on the drag term in the Fokker-Planck equation; in particular, the
drag term need not be corotational. With a square-integrable and
divergence-free initial velocity datum for the Navier-Stokes equation and a
nonnegative initial probability density function for the Fokker-Planck
equation, which has finite relative entropy with respect to the Maxwellian of
the model, we prove the existence of a global-in-time weak solution to the
coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the
absence of a body force, the weak solution decays exponentially in time to the
equilibrium solution, at a rate that is independent of the choice of the
initial datum and of the centre-of-mass diffusion coefficient.Comment: 86 page
Well posedness of an isothermal diffusive model for binary mixtures of incompressible fluids
We consider a model describing the behavior of a mixture of two
incompressible fluids with the same density in isothermal conditions. The model
consists of three balance equations: continuity equation, Navier-Stokes
equation for the mean velocity of the mixture, and diffusion equation
(Cahn-Hilliard equation). We assume that the chemical potential depends upon
the velocity of the mixture in such a way that an increase of the velocity
improves the miscibility of the mixture. We examine the thermodynamic
consistence of the model which leads to the introduction of an additional
constitutive force in the motion equation. Then, we prove existence and
uniqueness of the solution of the resulting differential problem
Compressible primitive equation: formal derivation and stability of weak solutions
We present a formal derivation of a simplified version of Compressible
Primitive Equations (CPEs) for atmosphere modeling. They are obtained from
-D compressible Navier-Stokes equations with an \emph{anisotropic viscous
stress tensor} where viscosity depends on the density. We then study the
stability of the weak solutions of this model by using an intermediate model,
called model problem, which is more simple and practical, to achieve the main
result
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