589 research outputs found
Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data
A simple proof of the existence of solutions for the two-dimensional
Keller-Segel model with measures with all the atoms less than as the
initial data is given. This result has been obtained by Senba--Suzuki and
Bedrossian--Masmoudi using different arguments. Moreover, we show a uniform
bound for the existence time of solutions as well as an optimal
hypercontractivity estimate
Non-uniqueness of weak solutions for the fractal Burgers equation
The notion of Kruzhkov entropy solution was extended by the first author in
2007 to conservation laws with a fractional laplacian diffusion term; this
notion led to well-posedness for the Cauchy problem in the
-framework. In the present paper, we further motivate the
introduction of entropy solutions, showing that in the case of fractional
diffusion of order strictly less than one, uniqueness of a weak solution may
fail.Comment: 23 page
Hydrodynamics in an external field
The methods of statistical dynamics are applied to a fluid with 5 conserved
fields (the mass, the energy, and the three components of momentum) moving in a
given external potential. When the potential is zero, we recover a previously
derived system of parabolic differential equations, called "corrections to
fluid dynamics".Comment: extends results of math-ph/0105013 in the presence of an external
field; to appear in Rep. Math. Phys. (2002
Global existence versus blow up for some models of interacting particles
We study the global existence and space-time asymptotics of solutions for a
class of nonlocal parabolic semilinear equations. Our models include the
Nernst-Planck and the Debye-Hukel drift-diffusion systems as well as
parabolic-elliptic systems of chemotaxis. In the case of a model of
self-gravitating particles, we also give a result on the finite time blow up of
solutions with localized and oscillating complex-valued initial data, using a
method by S. Montgomery-Smith.Comment: Colloq. Math. (to appear
Barenblatt profiles for a nonlocal porous media equation
We study a generalization of the porous medium equation involving nonlocal
terms. More precisely, explicit self-similar solutions with compact support
generalizing the Barenblatt solutions are constructed. We also present a formal
argument to get the decay of weak solutions of the corresponding Cauchy
problem.Comment: Note \`a para\^itre au Comptes-Rendus Math\'ematiqu
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