824 research outputs found
The 3-dimensional oscillon equation
On a bounded three-dimensional smooth domain, we consider the generalized
oscillon equation with Dirichlet boundary conditions, with time-dependent
damping and time-dependent squared speed of propagation. Under structural
assumptions on the damping and the speed of propagation, which include the
relevant physical case of reheating phase of inflation, we establish the
existence of a pullback global attractor of optimal regularity, and
finite-dimensionality of the kernel sections
Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations
Using the concept of stationary statistical solution, which generalizes the
notion of invariant measure, it is proved that, in a suitable sense, time
averages of almost every Leray-Hopf weak solution of the three-dimensional
incompressible Navier-Stokes equations converge as the averaging time goes to
infinity. This system of equations is not known to be globally well-posed, and
the above result answers a long-standing problem, extending to this system a
classical result from ergodic theory. It is also showed that, from a
measure-theoretic point of view, the stationary statistical solution obtained
from a generalized limit of time averages is independent of the choice of the
generalized limit. Finally, any Borel subset of the phase space with positive
measure with respect to a stationary statistical solution is such that for
almost all initial conditions in that Borel set and for at least one Leray-Hopf
weak solution starting with that initial condition, the corresponding orbit is
recurrent to that Borel subset and its mean sojourn time within that Borel
subset is strictly positive.Comment: Version 2: fixed some typos; added some references; and expanded some
sentences and some remarks for the sake of clarit
Time-Dependent Attractor for the Oscillon Equation
We investigate the asymptotic behavior of the nonautonomous evolution problem
generated by the Klein-Gordon equation in an expanding background, in one space
dimension with periodic boundary conditions, with a nonlinear potential of
arbitrary polynomial growth. After constructing a suitable dynamical framework
to deal with the explicit time dependence of the energy of the solution, we
establish the existence of a regular, time-dependent global attractor. The
sections of the attractor at given times have finite fractal dimension.Comment: to appear in Discrete and Continuous Dynamical System
Synchronization of extended systems from internal coherence
A condition for the synchronizability of a pair of PDE systems, coupled
through a finite set of variables, is commonly the existence of internal
synchronization or internal coherence in each system separately. The condition
was previously illustrated in a forced-dissipative system, and is here extended
to Hamiltonian systems, using an example from particle physics. Full
synchronization is precluded by Liouville's theorem. A form of synchronization
weaker than "measure synchronization" is manifest as the positional coincidence
of coherent oscillations ("breathers" or "oscillons") in a pair of coupled
scalar field models in an expanding universe with a nonlinear potential, and
does not occur with a variant of the model that does not exhibit oscillons.Comment: version accepted for publication in PRE (paragraph beginning at the
bottom of pg. 5 has been rewritten to suggest unifying principle for
synchronizability, applying to both forced-dissipative and Hamiltonian
systems; other minor changes
Discrete Data Assimilation in the Lorenz and 2D Navier--Stokes Equations
Consider a continuous dynamical system for which partial information about
its current state is observed at a sequence of discrete times. Discrete data
assimilation inserts these observational measurements of the reference
dynamical system into an approximate solution by means of an impulsive forcing.
In this way the approximating solution is coupled to the reference solution at
a discrete sequence of points in time. This paper studies discrete data
assimilation for the Lorenz equations and the incompressible two-dimensional
Navier--Stokes equations. In both cases we obtain bounds on the time interval h
between subsequent observations which guarantee the convergence of the
approximating solution obtained by discrete data assimilation to the reference
solution
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
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
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
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