43 research outputs found
Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,
This paper studies the transport of a mass in by a
flow field . We focus on kernels for
for which the smooth densities are known to develop
singularities in finite time. For this range This paper studies the transport
of a mass in by a flow field . We
focus on kernels for for which the
smooth densities are known to develop singularities in finite time. For this
range we prove the existence for all time of radially symmetric measure
solutions that are monotone decreasing as a function of the radius, thus
allowing for continuation of the solution past the blowup time. The monotone
constraint on the data is consistent with the typical blowup profiles observed
in recent numerical studies of these singularities. We prove monotonicity is
preserved for all time, even after blowup, in contrast to the case
where radially symmetric solutions are known to lose monotonicity. In the case
of the Newtonian potential (), under the assumption of radial
symmetry the equation can be transformed into the inviscid Burgers equation on
a half line. This enables us to prove preservation of monotonicity using the
classical theory of conservation laws. In the case and at
the critical exponent we exhibit initial data in for which the
solution immediately develops a Dirac mass singularity. This extends recent
work on the local ill-posedness of solutions at the critical exponent.Comment: 30 page
The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions
We study the multidimensional aggregation equation u_t+\Div(uv)=0,
with initial data in \cP_2(\bR^d)\cap L_{p}(\bR^d). We prove
that with biological relevant potential , the equation is ill-posed
in the critical Lebesgue space L_{d/(d-1)}(\bR^d) in the sense that there
exists initial data in \cP_2(\bR^d)\cap L_{d/(d-1)}(\bR^d) such that the
unique measure-valued solution leaves L_{d/(d-1)}(\bR^d) immediately. We also
extend this result to more general power-law kernels ,
for , and prove a conjecture in Bertozzi,
Laurent and Rosado [5] about instantaneous mass concentration for initial data
in \cP_2(\bR^d)\cap L_{p}(\bR^d) with . Finally, we classify all the
"first kind" radially symmetric similarity solutions in dimension greater than
two.Comment: typos corrected, 18 pages, to appear in Comm. Math. Phy
Equilibria of biological aggregations with nonlocal repulsive-attractive interactions
We consider the aggregation equation in , where the interaction potential
incorporates short-range Newtonian repulsion and long-range power-law
attraction. We study the global well-posedness of solutions and investigate
analytically and numerically the equilibrium solutions. We show that there
exist unique equilibria supported on a ball of . By using the
method of moving planes we prove that such equilibria are radially symmetric
and monotone in the radial coordinate. We perform asymptotic studies for the
limiting cases when the exponent of the power-law attraction approaches
infinity and a Newtonian singularity, respectively. Numerical simulations
suggest that equilibria studied here are global attractors for the dynamics of
the aggregation model
Aggregation-diffusion equations: dynamics, asymptotics, and singular limits
Given a large ensemble of interacting particles, driven by nonlocal
interactions and localized repulsion, the mean-field limit leads to a class of
nonlocal, nonlinear partial differential equations known as
aggregation-diffusion equations. Over the past fifteen years,
aggregation-diffusion equations have become widespread in biological
applications and have also attracted significant mathematical interest, due to
their competing forces at different length scales. These competing forces lead
to rich dynamics, including symmetrization, stabilization, and metastability,
as well as sharp dichotomies separating well-posedness from finite time blowup.
In the present work, we review known analytical results for
aggregation-diffusion equations and consider singular limits of these
equations, including the slow diffusion limit, which leads to the constrained
aggregation equation, as well as localized aggregation and vanishing diffusion
limits, which lead to metastability behavior. We also review the range of
numerical methods available for simulating solutions, with special attention
devoted to recent advances in deterministic particle methods. We close by
applying such a method -- the blob method for diffusion -- to showcase key
properties of the dynamics of aggregation-diffusion equations and related
singular limits
Nonlocal interactions by repulsive-attractive potentials: radial ins/stability
In this paper, we investigate nonlocal interaction equations with
repulsive-attractive radial potentials. Such equations describe the evolution
of a continuum density of particles in which they repulse each other in the
short range and attract each other in the long range. We prove that under some
conditions on the potential, radially symmetric solutions converge
exponentially fast in some transport distance toward a spherical shell
stationary state. Otherwise we prove that it is not possible for a radially
symmetric solution to converge weakly toward the spherical shell stationary
state. We also investigate under which condition it is possible for a
non-radially symmetric solution to converge toward a singular stationary state
supported on a general hypersurface. Finally we provide a detailed analysis of
the specific case of the repulsive-attractive power law potential as well as
numerical results. We point out the the conditions of radial ins/stability are
sharp.Comment: 42 pages, 7 figure
Swarm dynamics and equilibria for a nonlocal aggregation model
We consider the aggregation equation ρt − ∇ · (ρ∇K ∗ ρ) = 0 in Rn, where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of Rn and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric density ρ ̄ that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which ρ̄ is the steady state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results
The regularity of the boundary of a multidimensional aggregation patch
Let and let be the fundamental solution of the Laplace
equation in We consider the aggregation equation with
initial data , where is the indicator
function of a bounded domain We now fix and
take to be a bounded domain (a domain with smooth boundary
of class ). Then we have Theorem: If is a
domain, then the initial value problem above has a solution given by
where is a domain for all