857 research outputs found
Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
We analyze under which conditions equilibration between two competing
effects, repulsion modeled by nonlinear diffusion and attraction modeled by
nonlocal interaction, occurs. This balance leads to continuous compactly
supported radially decreasing equilibrium configurations for all masses. All
stationary states with suitable regularity are shown to be radially symmetric
by means of continuous Steiner symmetrization techniques. Calculus of
variations tools allow us to show the existence of global minimizers among
these equilibria. Finally, in the particular case of Newtonian interaction in
two dimensions they lead to uniqueness of equilibria for any given mass up to
translation and to the convergence of solutions of the associated nonlinear
aggregation-diffusion equations towards this unique equilibrium profile up to
translations as
Stationary States and Asymptotic Behaviour of Aggregation Models with Nonlinear Local Repulsion
We consider a continuum aggregation model with nonlinear local repulsion
given by a degenerate power-law diffusion with general exponent. The steady
states and their properties in one dimension are studied both analytically and
numerically, suggesting that the quadratic diffusion is a critical case. The
focus is on finite-size, monotone and compactly supported equilibria. We also
investigate numerically the long time asymptotics of the model by simulations
of the evolution equation. Issues such as metastability and local/ global
stability are studied in connection to the gradient flow formulation of the
model
A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
We propose a positivity preserving entropy decreasing finite volume scheme
for nonlinear nonlocal equations with a gradient flow structure. These
properties allow for accurate computations of stationary states and long-time
asymptotics demonstrated by suitably chosen test cases in which these features
of the scheme are essential. The proposed scheme is able to cope with
non-smooth stationary states, different time scales including metastability, as
well as concentrations and self-similar behavior induced by singular nonlocal
kernels. We use the scheme to explore properties of these equations beyond
their present theoretical knowledge
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