119 research outputs found
Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations
We determine an asymptotic expression of the blow-up time t_coll for
self-gravitating Brownian particles or bacterial populations (chemotaxis) close
to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with
t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian
particles) or the mass (for bacterial colonies), and eta_c is the critical
value of eta above which the system blows up. This result is in perfect
agreement with the numerical solution of the Smoluchowski-Poisson system. We
also determine the asymptotic expression of the relaxation time close but above
the critical temperature and derive a large time asymptotic expansion for the
density profile exactly at the critical point
Critical dynamics of self-gravitating Langevin particles and bacterial populations
We study the critical dynamics of the generalized Smoluchowski-Poisson system
(for self-gravitating Langevin particles) or generalized Keller-Segel model
(for the chemotaxis of bacterial populations). These models [Chavanis & Sire,
PRE, 69, 016116 (2004)] are based on generalized stochastic processes leading
to the Tsallis statistics. The equilibrium states correspond to polytropic
configurations with index similar to polytropic stars in astrophysics. At
the critical index (where is the dimension of space),
there exists a critical temperature (for a given mass) or a
critical mass (for a given temperature). For or
the system tends to an incomplete polytrope confined by the box (in a
bounded domain) or evaporates (in an unbounded domain). For
or the system collapses and forms, in a finite time, a Dirac peak
containing a finite fraction of the total mass surrounded by a halo. This
study extends the critical dynamics of the ordinary Smoluchowski-Poisson system
and Keller-Segel model in corresponding to isothermal configurations with
. We also stress the analogy between the limiting mass of
white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial
populations in the generalized Keller-Segel model of chemotaxis
Continuous dependence estimates for nonlinear fractional convection-diffusion equations
We develop a general framework for finding error estimates for
convection-diffusion equations with nonlocal, nonlinear, and possibly
degenerate diffusion terms. The equations are nonlocal because they involve
fractional diffusion operators that are generators of pure jump Levy processes
(e.g. the fractional Laplacian). As an application, we derive continuous
dependence estimates on the nonlinearities and on the Levy measure of the
diffusion term. Estimates of the rates of convergence for general nonlinear
nonlocal vanishing viscosity approximations of scalar conservation laws then
follow as a corollary. Our results both cover, and extend to new equations, a
large part of the known error estimates in the literature.Comment: In this version we have corrected Example 3.4 explaining the link
with the results in [51,59
Some Applications of Fractional Equations
We present two observations related to theapplication of linear (LFE) and
nonlinear fractional equations (NFE). First, we give the comparison and
estimates of the role of the fractional derivative term to the normal diffusion
term in a LFE. The transition of the solution from normal to anomalous
transport is demonstrated and the dominant role of the power tails in the long
time asymptotics is shown. Second, wave propagation or kinetics in a nonlinear
media with fractal properties is considered. A corresponding fractional
generalization of the Ginzburg-Landau and nonlinear Schrodinger equations is
proposed.Comment: 11 page
Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2
In this paper we prove finite-time blowup of radially symmetric solutions to
the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any
positive mass. This is done in case of nonlinear diffusion and also in the case
of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is
assumed not to decay. Moreover, it is shown that the above-mentioned lack of
non-decay assumption is essential with respect to keeping the dichotomy
finite-time blowup against boundedness of solutions. Namely, we prove that
without the non-decay assumption possible asymptotic behaviour of solutions
includes also infinite-time blowup.Comment: 14 page
Competence and Professional Skills in Training Future Specialists in the Field of Physical Education and Sports
The relevance of the problem is determined by the need to understand the further strategy of improving the training of physical education and sports specialists. The purpose of the article is to study the problem of correlation between the notions of âcompetenceâ and âprofessional skillâ in the context of the formation of a specialist in the field of physical education and sports. The study of the relationship between the notions of âcompetenceâ and âprofessional skillâ was conducted on the basis of an array of special literature on the competent approach to training of physical education and sports specialists, including more than 200 sources published in different countries. The article analyses negative trends in the content of training future specialists in the system of higher professional physical education based on a competence approach. The acme-culturological approach for improving the professional training of future specialists is proposed and tested, the stages of formation of professional skills of future specialists in the training process are presented. There is a need to focus on the identified features of professional training of young people to professions in the field of physical education and sports. The proposed system of professional training of future specialists of physical education and sports on acme-culturological bases directs teachers to improve the quality of training graduates in their future professional activities
Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions
We address the generalized thermodynamics and the collapse of a system of
self-gravitating Langevin particles exhibiting anomalous diffusion in a space
of dimension D. The equilibrium states correspond to polytropic distributions.
The index n of the polytrope is related to the exponent of anomalous diffusion.
We consider a high-friction limit and reduce the problem to the study of the
nonlinear Smoluchowski-Poisson system. We show that the associated Lyapunov
functional is the Tsallis free energy. We discuss in detail the equilibrium
phase diagram of self-gravitating polytropes as a function of D and n and
determine their stability by using turning points arguments and analytical
methods. When no equilibrium state exists, we investigate self-similar
solutions describing the collapse. These results can be relevant for
astrophysical systems, two-dimensional vortices and for the chemotaxis of
bacterial populations. Above all, this model constitutes a prototypical
dynamical model of systems with long-range interactions which possesses a rich
structure and which can be studied in great detail.Comment: Submitted to Phys. Rev.
Thermodynamics of self-gravitating systems
Self-gravitating systems are expected to reach a statistical equilibrium
state either through collisional relaxation or violent collisionless
relaxation. However, a maximum entropy state does not always exist and the
system may undergo a ``gravothermal catastrophe'': it can achieve ever
increasing values of entropy by developing a dense and hot ``core'' surrounded
by a low density ``halo''. In this paper, we study the phase transition between
``equilibrium'' states and ``collapsed'' states with the aid of a simple
relaxation equation [Chavanis, Sommeria and Robert, Astrophys. J. 471, 385
(1996)] constructed so as to increase entropy with an optimal rate while
conserving mass and energy. With this numerical algorithm, we can cover the
whole bifurcation diagram in parameter space and check, by an independent
method, the stability limits of Katz [Mon. Not. R. astr. Soc. 183, 765 (1978)]
and Padmanabhan [Astrophys. J. Supp. 71, 651 (1989)]. When no equilibrium state
exists, our relaxation equation develops a self-similar collapse leading to a
finite time singularity.Comment: 54 pages. 25 figures. Submitted to Phys. Rev.
A study of blow-ups in the Keller-Segel model of chemotaxis
We study the Keller-Segel model of chemotaxis and develop a composite
particle-grid numerical method with adaptive time stepping which allows us to
accurately resolve singular solutions. The numerical findings (in two
dimensions) are then compared with analytical predictions regarding formation
and interaction of singularities obtained via analysis of the stochastic
differential equations associated with the Keller-Segel model
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