197 research outputs found
Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold
We consider the asymptotic behaviour of positive solutions of the
fast diffusion equation
posed for x\in\RR^d, , with a precise value for the exponent
. The space dimension is so that , and even
for . This case had been left open in the general study \cite{BBDGV} since
it requires quite different functional analytic methods, due in particular to
the absence of a spectral gap for the operator generating the linearized
evolution.
The linearization of this flow is interpreted here as the heat flow of the
Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}),
with a metric which is conformal to the standard \RR^d metric.
Studying the pointwise heat kernel behaviour allows to prove {suitable
Gagliardo-Nirenberg} inequalities associated to the generator. Such
inequalities in turn allow to study the nonlinear evolution as well, and to
determine its asymptotics, which is identical to the one satisfied by the
linearization. In terms of the rescaled representation, which is a nonlinear
Fokker--Planck equation, the convergence rate turns out to be polynomial in
time. This result is in contrast with the known exponential decay of such
representation for all other values of .Comment: 37 page
Eternal solutions to a singular diffusion equation with critical gradient absorption
The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type is investigated for the singular diffusion equation with critical gradient absorption \begin{equation*} \partial_{t} u-\Delta_{p} u+|\nabla u|^{p/2}=0 \quad \;\;\hbox{in}\;\; (0,\infty)\times\real^N \end{equation*} where . Such solutions are shown to exist only if the parameter ranges in a bounded interval which is in sharp contrast with well-known singular diffusion equations such as when or the porous medium equation when . Moreover, the profile decays to zero as in a faster way for than for but the algebraic leading order is the same in both cases. In fact, for large , decays as while behaves as when
Existence of Ricci flows of incomplete surfaces
We prove a general existence result for instantaneously complete Ricci flows
starting at an arbitrary Riemannian surface which may be incomplete and may
have unbounded curvature. We give an explicit formula for the maximal existence
time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction
Formal matched asymptotics for degenerate Ricci flow neckpinches
Gu and Zhu have shown that Type-II Ricci flow singularities develop from
nongeneric rotationally symmetric Riemannian metrics on , for all . In this paper, we describe and provide plausibility arguments for a
detailed asymptotic profile and rate of curvature blow-up that we predict such
solutions exhibit
Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation
We study the asymptotic behaviour of positive solutions of the Cauchy problem
for the fast diffusion equation near the extinction time. We find a continuum
of rates of convergence to a self-similar profile. These rates depend
explicitly on the spatial decay rates of initial data
From wellness to medical diagnostic apps: the Parkinson's Disease case
This paper presents the design and development of the CloudUPDRS app and supporting system developed as a Class I medical device to assess the severity of motor symptoms for Parkinsonâs Disease. We report on lessons learnt towards meeting fidelity and regulatory requirements; effective procedures employed to structure user context and ensure data quality; a robust service provision architecture; a dependable analytics toolkit; and provisions to meet mobility and social needs of people with Parkinsonâs
Poland's syndrome and recurrent pneumothorax: is there a connection?
Aim. To investigate the possible connection of Poland's syndrome with the presence of lung bullae and, thus, with an increased risk for recurrent pneumothorax. Patients-methods. Two male patients, aged 19 and 21 years respectively were submitted to our department after their second incident of pneumothorax. Both had Poland's syndrome (unilaterally hypoplastic chest wall with pectoralis major muscle atrophy) and both had multiple bullae to the ipsilateral lung based on CT findings. The patients were treated operatively (bullectomy, lung apicectomy, partial parietal pleurectomy and chemical pleurodesis) due to the recurrent state of their pneumothorax. Results. The patients had good results with total expansion of the affected lung. Conclusions. Poland's syndrome can be combined with ipsilateral presence of lung bullae, a common cause of pneumothorax. Whether this finding is part or a variation of the syndrome needs to be confirmed by a larger number of similar cases
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