366 research outputs found
The dynamics of the 3D radial NLS with the combined terms
In this paper, we show the scattering and blow-up result of the radial
solution with the energy below the threshold for the nonlinear Schr\"{o}dinger
equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u
\tag{CNLS} in the energy space . The threshold is given by the
ground state for the energy-critical NLS: . This
problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The
main difficulty is the lack of the scaling invariance. Illuminated by
\cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the
scaling parameter, then apply it into the scattering theory. Our result shows
that the defocusing, -subcritical perturbation does not
affect the determination of the threshold of the scattering solution of (CNLS)
in the energy space.Comment: 46page
Nondispersive solutions to the L2-critical half-wave equation
We consider the focusing -critical half-wave equation in one space
dimension where denotes the
first-order fractional derivative. Standard arguments show that there is a
critical threshold such that all solutions with extend globally in time, while solutions with may develop singularities in finite time.
In this paper, we first prove the existence of a family of traveling waves
with subcritical arbitrarily small mass. We then give a second example of
nondispersive dynamics and show the existence of finite-time blowup solutions
with minimal mass . More precisely, we construct a
family of minimal mass blowup solutions that are parametrized by the energy
and the linear momentum . In particular, our main result
(and its proof) can be seen as a model scenario of minimal mass blowup for
-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page
Semiclassical approximations for Hamiltonians with operator-valued symbols
We consider the semiclassical limit of quantum systems with a Hamiltonian
given by the Weyl quantization of an operator valued symbol. Systems composed
of slow and fast degrees of freedom are of this form. Typically a small
dimensionless parameter controls the separation of time
scales and the limit corresponds to an adiabatic limit, in
which the slow and fast degrees of freedom decouple. At the same time
is the semiclassical limit for the slow degrees of freedom.
In this paper we show that the -dependent classical flow for the
slow degrees of freedom first discovered by Littlejohn and Flynn, coming from
an \epsi-dependent classical Hamilton function and an -dependent
symplectic form, has a concrete mathematical and physical meaning: Based on
this flow we prove a formula for equilibrium expectations, an Egorov theorem
and transport of Wigner functions, thereby approximating properties of the
quantum system up to errors of order . In the context of Bloch
electrons formal use of this classical system has triggered considerable
progress in solid state physics. Hence we discuss in some detail the
application of the general results to the Hofstadter model, which describes a
two-dimensional gas of non-interacting electrons in a constant magnetic field
in the tight-binding approximation.Comment: Final version to appear in Commun. Math. Phys. Results have been
strengthened with only minor changes to the proofs. A section on the
Hofstadter model as an application of the general theory was added and the
previous section on other applications was remove
Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption
We extend our previous result on the focusing cubic Klein-Gordon equation in
three dimensions to the non-radial case, giving a complete classification of
global dynamics of all solutions with energy at most slightly above that of the
ground state.Comment: 40 page
Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation
We consider the mass-critical focusing nonlinear Schrodinger equation in the
presence of an external potential, when the nonlinearity is inhomogeneous. We
show that if the inhomogeneous factor in front of the nonlinearity is
sufficiently flat at a critical point, then there exists a solution which blows
up in finite time with the maximal (unstable) rate at this point. In the case
where the critical point is a maximum, this solution has minimal mass among the
blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of
the mass-critical Schrodinger equation on some surfaces. The proof is based on
properties of the linearized operator around the ground state, and on a full
use of the invariances of the equation with an homogeneous nonlinearity and no
potential, via time-dependent modulations.Comment: 36 pages. More explanations, references updated, statement of Theorem
1.1 corrected. FInal versio
Protective effects of angiopoietin-like 4 on cerebrovascular and functional damages in ischaemic stroke
AIMS: Given the impact of vascular injuries and oedema on brain damage caused during stroke, vascular protection represents a major medical need. We hypothesized that angiopoietin-like 4 (ANGPTL4), a regulator of endothelial barrier integrity, might exert a protective effect during ischaemic stroke. METHODS AND RESULTS: Using a murine transient ischaemic stroke model, treatment with recombinant ANGPTL4 led to significantly decreased infarct size and improved behaviour. Quantitative characteristics of the vascular network (density and branchpoints) were preserved in ANGPTL4-treated mice. Integrity of tight and adherens junctions was also quantified and ANGPTL4-treated mice displayed increased VE-cadherin and claudin-5-positive areas. Brain oedema was thus significantly decreased in ANGPTL4-treated mice. In accordance, vascular damage and infarct severity were increased in angptl4-deficient mice thus providing genetic evidence that ANGPTL4 preserves brain tissue from ischaemia-induced alterations. Altogether, these data show that ANGPTL4 protects not only the global vascular network, but also interendothelial junctions and controls both deleterious inflammatory response and oedema. Mechanistically, ANGPTL4 counteracted VEGF signalling and thereby diminished Src-signalling downstream from VEGFR2. This led to decreased VEGFR2-VE-cadherin complex disruption, increased stability of junctions and thus increased endothelial cell barrier integrity of the cerebral microcirculation. In addition, ANGPTL4 prevented neuronal loss in the ischaemic area. CONCLUSION: These results, therefore, show ANGPTL4 counteracts the loss of vascular integrity in ischaemic stroke, by restricting Src kinase signalling downstream from VEGFR2. ANGPTL4 treatment thus reduces oedema, infarct size, neuronal loss, and improves mice behaviour. These results suggest that ANGPTL4 constitutes a relevant target for vasculoprotection and cerebral protection during stroke
On Polynomial Stability of Coupled Partial Differential Equations in 1D
We study the well-posedness and asymptotic behaviour of selected PDE-PDE and
PDE-ODE systems on one-dimensional spatial domains, namely a boundary coupled
wave-heat system and a wave equation with a dynamic boundary condition. We
prove well-posedness of the models and derive rational decay rates for the
energy using an approach where the coupled systems are formulated as feedback
interconnections of impedance passive regular linear systems.Comment: 12 pages, 1 figure, accepted for publication in the Proceedings of
"Semigroups of Operators: Theory and Applications", Kazimierz Dolny, Poland,
October 201
Tau pathology in aged cynomolgus monkeys is progressive supranuclear palsy/corticobasal degeneration- but not Alzheimer disease-like -Ultrastructural mapping of tau by EDX-
Silver diagnosis in neuropathology: principles, practice and revised interpretation
Silver-staining methods are helpful for histological identification of pathological deposits. In spite of some ambiguities regarding their mechanism and interpretation, they are widely used for histopathological diagnosis. In this review, four major silver-staining methods, modified Bielschowsky, Bodian, Gallyas (GAL) and Campbell–Switzer (CS) methods, are outlined with respect to their principles, basic protocols and interpretations, thereby providing neuropathologists, technicians and neuroscientists with a common basis for comparing findings and identifying the issues that still need to be clarified. Some consider “argyrophilia” to be a homogeneous phenomenon irrespective of the lesion and the method. Thus, they seek to explain the differences among the methods by pointing to their different sensitivities in detecting lesions (quantitative difference). Comparative studies, however, have demonstrated that argyrophilia is heterogeneous and dependent not only on the method but also on the lesion (qualitative difference). Each staining method has its own lesion-dependent specificity and, within this specificity, its own sensitivity. This “method- and lesion-dependent” nature of argyrophilia enables operational sorting of disease-specific lesions based on their silver-staining profiles, which may potentially represent some disease-specific aspects. Furthermore, comparisons between immunohistochemical and biochemical data have revealed an empirical correlation between GAL+/CS-deposits and 4-repeat (4R) tau (corticobasal degeneration, progressive supranuclear palsy and argyrophilic grains) and its complementary reversal between GAL-/CS+deposits and 3-repeat (3R) tau (Pick bodies). Deposits containing both 3R and 4R tau (neurofibrillary tangles of Alzheimer type) are GAL+/CS+. Although no molecular explanations, other than these empiric correlations, are currently available, these distinctive features, especially when combined with immunohistochemistry, are useful because silver-staining methods and immunoreactions are complementary to each other
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