708 research outputs found
Global exponential stability of classical solutions to the hydrodynamic model for semiconductors
In this paper, the global well-posedness and stability of classical solutions
to the multidimensional hydrodynamic model for semiconductors on the framework
of Besov space are considered. We weaken the regularity requirement of the
initial data, and improve some known results in Sobolev space. The local
existence of classical solutions to the Cauchy problem is obtained by the
regularized means and compactness argument. Using the high- and low- frequency
decomposition method, we prove the global exponential stability of classical
solutions (close to equilibrium). Furthermore, it is also shown that the
vorticity decays to zero exponentially in the 2D and 3D space. The main
analytic tools are the Littlewood-Paley decomposition and Bony's para-product
formula.Comment: 18 page
Blow-up of the hyperbolic Burgers equation
The memory effects on microscopic kinetic systems have been sometimes
modelled by means of the introduction of second order time derivatives in the
macroscopic hydrodynamic equations. One prototypical example is the hyperbolic
modification of the Burgers equation, that has been introduced to clarify the
interplay of hyperbolicity and nonlinear hydrodynamic evolution. Previous
studies suggested the finite time blow-up of this equation, and here we present
a rigorous proof of this fact
Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System
We study the Hilbert expansion for small Knudsen number for the
Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term
takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{\rho_{0}(t,x)}{(2\pi
\theta_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2\theta_{0}(t,x)},\text{\
}\theta_{0}(t,x)=K\rho_{0}^{2/3}(t,x).t=0u_00\leq t\leq \varepsilon
^{-{1/2}\frac{2k-3}{2k-2}},\rho_{0}(t,x) u_{0}(t,x)\gamma=5/3$
Development of singularities for the compressible Euler equations with external force in several dimensions
We consider solutions to the Euler equations in the whole space from a
certain class, which can be characterized, in particular, by finiteness of
mass, total energy and momentum. We prove that for a large class of right-hand
sides, including the viscous term, such solutions, no matter how smooth
initially, develop a singularity within a finite time. We find a sufficient
condition for the singularity formation, "the best sufficient condition", in
the sense that one can explicitly construct a global in time smooth solution
for which this condition is not satisfied "arbitrary little". Also compactly
supported perturbation of nontrivial constant state is considered. We
generalize the known theorem by Sideris on initial data resulting in
singularities. Finally, we investigate the influence of frictional damping and
rotation on the singularity formation.Comment: 23 page
Dispersion and collapse of wave maps
We study numerically the Cauchy problem for equivariant wave maps from 3+1
Minkowski spacetime into the 3-sphere. On the basis of numerical evidence
combined with stability analysis of self-similar solutions we formulate two
conjectures. The first conjecture states that singularities which are produced
in the evolution of sufficiently large initial data are approached in a
universal manner given by the profile of a stable self-similar solution. The
second conjecture states that the codimension-one stable manifold of a
self-similar solution with exactly one instability determines the threshold of
singularity formation for a large class of initial data. Our results can be
considered as a toy-model for some aspects of the critical behavior in
formation of black holes.Comment: 14 pages, Latex, 9 eps figures included, typos correcte
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-
Black Hole Critical Phenomena Without Black Holes
Studying the threshold of black hole formation via numerical evolution has
led to the discovery of fascinating nonlinear phenomena. Power-law mass
scaling, aspects of universality, and self-similarity have now been found for a
large variety of models. However, questions remain. Here I briefly review
critical phenomena, discuss some recent results, and describe a model which
demonstrates similar phenomena without gravity.Comment: 13 pages, 6 figures; Submission for the proceedings of ICGC 2000 in
the journal Preman
Dynamical elastic bodies in Newtonian gravity
Well-posedness for the initial value problem for a self-gravitating elastic
body with free boundary in Newtonian gravity is proved. In the material frame,
the Euler-Lagrange equation becomes, assuming suitable constitutive properties
for the elastic material, a fully non-linear elliptic-hyperbolic system with
boundary conditions of Neumann type. For systems of this type, the initial data
must satisfy compatibility conditions in order to achieve regular solutions.
Given a relaxed reference configuration and a sufficiently small Newton's
constant, a neigborhood of initial data satisfying the compatibility conditions
is constructed
Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model
We study the distinction and quantification of chaotic and regular motion in
a time-dependent Hamiltonian barred galaxy model. Recently, a strong
correlation was found between the strength of the bar and the presence of
chaotic motion in this system, as models with relatively strong bars were shown
to exhibit stronger chaotic behavior compared to those having a weaker bar
component. Here, we attempt to further explore this connection by studying the
interplay between chaotic and regular behavior of star orbits when the
parameters of the model evolve in time. This happens for example when one
introduces linear time dependence in the mass parameters of the model to mimic,
in some general sense, the effect of self-consistent interactions of the actual
N-body problem. We thus observe, in this simple time-dependent model also, that
the increase of the bar's mass leads to an increase of the system's chaoticity.
We propose a new way of using the Generalized Alignment Index (GALI) method as
a reliable criterion to estimate the relative fraction of chaotic vs. regular
orbits in such time-dependent potentials, which proves to be much more
efficient than the computation of Lyapunov exponents. In particular, GALI is
able to capture subtle changes in the nature of an orbit (or ensemble of
orbits) even for relatively small time intervals, which makes it ideal for
detecting dynamical transitions in time-dependent systems.Comment: 21 pages, 9 figures (minor typos fixed) to appear in J. Phys. A:
Math. Theo
Concerning the Wave equation on Asymptotically Euclidean Manifolds
We obtain KSS, Strichartz and certain weighted Strichartz estimate for the
wave equation on , , when metric
is non-trapping and approaches the Euclidean metric like with
. Using the KSS estimate, we prove almost global existence for
quadratically semilinear wave equations with small initial data for
and . Also, we establish the Strauss conjecture when the metric is radial
with for .Comment: Final version. To appear in Journal d'Analyse Mathematiqu
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