7 papers
Degenerate constants in degree inequalities for Sobolev circle maps: on some problems posed by Brezis
Xu'an Dou, Zeyu Jin
We sharpen the constants in two degree inequalities for circle-valued Sobolev maps in degenerate regimes, as or . The two proofs use the same power trick toge…
Numerical analysis for leaky-integrate-fire networks under Euler-Maruyama
Xu'an Dou, Frank Chen, Kevin K Lin +1
Leaky integrate-and-fire (LIF) networks are standard reduced models for spike-based neural dynamics and a natural substrate for neuromorphic computation. We study time-driven Euler…
Resolving the Blow-Up: A Time-Dilated Numerical Framework for Multiple Firing Events in Mean-Field Neuronal Networks
Xu'an Dou, Louis Tao, Zhe Xue +1
In large-scale excitatory neuronal networks, rapid synchronization manifests as {multiple firing events (MFEs)}, mathematically characterized by a finite-time blow-up of the neuron…
On partial diffusion and mixing without hypoellipticity
Xu'an Dou, Delphine Salort, Didier Smets
A simple Markov process is considered involving a diffusion in one direction and a transport in a transverse direction. Quantitative mixing rate estimates are obtained with limited…
Noisy integrate-and-fire equation: continuation after blow-up
Xu'An Dou, Benoît Perthame, Delphine Salort +1
The integrate and fire equation is a classical model for neural assemblies which can exhibit finite time blow-up. A major open problem is to understand how to continue solutions af…
Classical solutions of a mean field system for pulse-coupled oscillators: long time asymptotics versus blowup
José Antonio Carrillo, Xu'an Dou, Pierre Roux +1
We introduce a novel reformulation of the mean-field system for pulse-coupled oscillators. It is based on writing a closed equation for the inverse distribution function associated…
Tumor growth with a necrotic core as an obstacle problem in pressure
Xu'an Dou, Chengfeng Shen, Zhennan Zhou
Motivated by the incompressible limit of a cell density model, we propose a free boundary tumor growth model where the pressure satisfies an obstacle problem on an evolving domain…