3,444 research outputs found
Hardy-Littlewood-Sobolev Inequalities via Fast Diffusion Flows
We give a simple proof of the cases of the sharp
Hardy-Littlewood-Sobolev inequality for , and the sharp Logarithmic
Hardy-Littlewood-Sobolev inequality for via a monotone flow governed by
the fast diffusion equation
Condensation phenomena in nonlinear drift equations
We study nonnegative, measure-valued solutions to nonlinear drift type
equations modelling concentration phenomena related to Bose-Einstein particles.
In one spatial dimension, we prove existence and uniqueness for measure
solutions. Moreover, we prove that all solutions blow up in finite time leading
to a concentration of mass only at the origin, and the concentrated mass
absorbs increasingly the mass converging to the total mass as time goes to
infinity. Our analysis makes a substantial use of independent variable scalings
and pseudo-inverse functions techniques
A particle method for the homogeneous Landau equation
We propose a novel deterministic particle method to numerically approximate
the Landau equation for plasmas. Based on a new variational formulation in
terms of gradient flows of the Landau equation, we regularize the collision
operator to make sense of the particle solutions. These particle solutions
solve a large coupled ODE system that retains all the important properties of
the Landau operator, namely the conservation of mass, momentum and energy, and
the decay of entropy. We illustrate our new method by showing its performance
in several test cases including the physically relevant case of the Coulomb
interaction. The comparison to the exact solution and the spectral method is
strikingly good maintaining 2nd order accuracy. Moreover, an efficient
implementation of the method via the treecode is explored. This gives a proof
of concept for the practical use of our method when coupled with the classical
PIC method for the Vlasov equation.Comment: 27 pages, 14 figures, debloated some figures, improved explanations
in sections 2, 3, and
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