Asymptotics of blowup solutions for the aggregation equation
Yanghong Huang Andrea Bertozzi
Discrete & Continuous Dynamical Systems - B 2012, 17(4): 1309-1331 doi: 10.3934/dcdsb.2012.17.1309
We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation $ u_t = \nabla\cdot(u\nabla K*u) $ in $\mathbb{R}^n$, for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$. For $\gamma>2$, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing $\delta$-ring. We develop an asymptotic theory for the approach to this singular solution. For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all $\gamma$ in this range, including additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.
keywords: asymptotic behavior self-similar solutions. blowup Aggregation equation
Explicit equilibrium solutions for the aggregation equation with power-law potentials
José A. Carrillo Yanghong Huang
Kinetic & Related Models 2017, 10(1): 171-192 doi: 10.3934/krm.2017007

Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernelsare constructed by inverting Fredholm integral operators or byemploying certain integral identities. These solutions are expected tobe the global energy stable equilibria and to characterize the generic behaviorsof stationary solutions for more general interactions.

keywords: Fredholm integral equations stationary solutions
Numerical study of a particle method for gradient flows
José Antonio Carrillo Yanghong Huang Francesco Saverio Patacchini Gershon Wolansky
Kinetic & Related Models 2017, 10(3): 613-641 doi: 10.3934/krm.2017025

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

keywords: Particle method diffusion aggregation gradient flow discrete gradient flow JKO scheme

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