2,586 research outputs found
Improved intermediate asymptotics for the heat equation
International audienceThis letter is devoted to results on intermediate asymptotics for the heat equation. We study the convergence towards a stationary solution in self-similar variables. By assuming the equality of some moments of the initial data and of the stationary solution, we get improved convergence rates using entropy / entropy-production methods. We establish the equivalence of the exponential decay of the entropies with new, improved functional inequalities in restricted classes of functions. This letter is the counterpart in a linear framework of a recent work on fast diffusion equations, see [Bonforte-Dolbeault-Grillo-Vazquez]. Results extend to the case of a Fokker-Planck equation with a general confining potential
Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations
We investigate the large-time asymptotics of nonlinear diffusion equations
in dimension , in the exponent interval , when the initial datum is of bounded second moment. Precise
rates of convergence to the Barenblatt profile in terms of the relative R\'enyi
entropy are demonstrated for finite-mass solutions defined in the whole space
when they are re-normalized at each time with respect to their own
second moment. The analysis shows that the relative R\'enyi entropy exhibits a
better decay, for intermediate times, with respect to the standard
Ralston-Newton entropy. The result follows by a suitable use of the so-called
concavity of R\'enyi entropy power
Best matching Barenblatt profiles are delayed
The growth of the second moments of the solutions of fast diffusion equations
is asymptotically governed by the behavior of self-similar solutions. However,
at next order, there is a correction term which amounts to a delay depending on
the nonlinearity and on a distance of the initial data to the set of
self-similar Barenblatt solutions. This distance can be measured in terms of a
relative entropy to the best matching Barenblatt profile. This best matching
Barenblatt function determines a scale. In new variables based on this scale,
which are given by a self-similar change of variables if and only if the
initial datum is one of the Barenblatt profiles, the typical scale is monotone
and has a l
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
Quantum corrections to the noncommutative kink
We calculate quantum corrections to the mass of noncommutative phi^4 kink in
(1+1) dimensions for intermediate and large values of the noncommutativity
parameter theta. All one-loop divergences are removed by a mass renormalization
(which is different from the one required in the topologically trivial sector).
For large theta quantum corrections to the mass grow linearly with theta
signaling about possible break down of the perturbative expansion.Comment: 18 pages, v2: minor change
Large-time Behavior of the Solutions to Rosenau Type Approximations to the Heat Equation
In this paper we study the large-time behavior of the solution to a general
Rosenau type approximation to the heat equation, by showing that the solution
to this approximation approaches the fundamental solution of the heat equation
at a sub-optimal rate. The result is valid in particular for the central
differences scheme approximation of the heat equation, a property which to our
knowledge has never been observed before.Comment: 20 page
- …