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On recurrence for $\mathbb{Z}^d$-nilsystems
Sebasti\'an Donoso, Felipe Hern\'andez, Alejandro Maass
https://arxiv.org/abs/2402.16838 https://

On recurrence for $\mathbb{Z}^d$-nilsystems
We study the topological recurrence phenomenon of actions of locally compact abelian groups on compact metric spaces. We address the popular Katznelson's question in this context and extend recurrence lifting results, known for $\mathbb{Z}$-actions, to this more general context. In the case $H=\mathbb{Z}^d$, we develop new techniques to analyze Bohr recurrence sets, and show that Bohr recurrence implies recurrence for a $\mathbb{Z}^d$-nilsystem $(G/Γ,\mathbb{Z}^d)$, under the assumption that t…

Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming
Nilasis Chaudhuri, Young-Pil Choi, Oliver Tse, Ewelina Zatorska
https://arxiv.org/abs/2402.07130

Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming
We consider a one-dimensional hydrodynamic model featuring nonlocal attraction-repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and the corresponding energy-type inequality, in the spirit of the Bresch-Desjardins estimate. We identify a dependence between the communication weight and interaction kernel and between the pressure and viscosity term allowing for this inequality to be uniform in time. It is then used to study long-time asymptotics of …