700 research outputs found
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
Global Solutions for Incompressible Viscoelastic Fluids
We prove the existence of both local and global smooth solutions to the
Cauchy problem in the whole space and the periodic problem in the n-dimensional
torus for the incompressible viscoelastic system of Oldroyd-B type in the case
of near equilibrium initial data. The results hold in both two and three
dimensional spaces. The results and methods presented in this paper are also
valid for a wide range of elastic complex fluids, such as magnetohydrodynamics,
liquid crystals and mixture problems.Comment: We prove the existence of global smooth solutions to the Cauchy
problem for the incompressible viscoelastic system of Oldroyd-B type in the
case of near equilibrium initial dat
Chaotic Orbits in Thermal-Equilibrium Beams: Existence and Dynamical Implications
Phase mixing of chaotic orbits exponentially distributes these orbits through
their accessible phase space. This phenomenon, commonly called ``chaotic
mixing'', stands in marked contrast to phase mixing of regular orbits which
proceeds as a power law in time. It is operationally irreversible; hence, its
associated e-folding time scale sets a condition on any process envisioned for
emittance compensation. A key question is whether beams can support chaotic
orbits, and if so, under what conditions? We numerically investigate the
parameter space of three-dimensional thermal-equilibrium beams with space
charge, confined by linear external focusing forces, to determine whether the
associated potentials support chaotic orbits. We find that a large subset of
the parameter space does support chaos and, in turn, chaotic mixing. Details
and implications are enumerated.Comment: 39 pages, including 14 figure
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
Geometric optics and instability for semi-classical Schrodinger equations
We prove some instability phenomena for semi-classical (linear or) nonlinear
Schrodinger equations. For some perturbations of the data, we show that for
very small times, we can neglect the Laplacian, and the mechanism is the same
as for the corresponding ordinary differential equation. Our approach allows
smaller perturbations of the data, where the instability occurs for times such
that the problem cannot be reduced to the study of an o.d.e.Comment: 22 pages. Corollary 1.7 adde
Soluble amyloid beta-containing aggregates are present throughout the brain at early stages of Alzheimer's disease.
Protein aggregation likely plays a key role in the initiation and spreading of Alzheimer's disease pathology through the brain. Soluble aggregates of amyloid beta are believed to play a key role in this process. However, the aggregates present in humans are still poorly characterized due to a lack of suitable methods required for characterizing the low concentration of heterogeneous aggregates present. We have used a variety of biophysical methods to characterize the aggregates present in human Alzheimer's disease brains at Braak stage III. We find soluble amyloid beta-containing aggregates in all regions of the brain up to 200 nm in length, capable of causing an inflammatory response. Rather than aggregates spreading through the brain as disease progresses, it appears that aggregation occurs all over the brain and that different brain regions are at earlier or later stages of the same process, with the later stages causing increased inflammation
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