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Federico Buseghin

Publications and source records attributed to Federico Buseghin.

9 recordsLinked to original sources
arXiv2026

Determination of the long-time dynamics for the 2D Keller-Segel equation at critical mass

We consider the parabolic-elliptic Keller-Segel equation in two dimensions on the whole space. We prove that for arbitrary initial data with critical mass $8\pi$ and finite second momentum, all solutions have the same universal behaviour. They are globally defined and, asymptotically for large times, they converge to a renormalized stationary state of the equation which concentrates around the center of mass of the solution at a universal logarithmic-in-time scale. Our result holds for general solutions without symmetry assumptions, and we furthermore provide explicit convergence rates. In the radial case, by combining our result with previous ones (by Blanchet-Dolbeault-Perthame, Mizoguchi, and related works), this achieves a complete classification of the possible dynamics: for subcritical masses solutions converge to a self-similar expander, at the critical mass they concentrate a stationary state in infinite time at the aforementioned universal scale, and for supercritical masses they blow up in finite time by type II concentration of a stationary state at another universal scale. Our proof starts with soliton resolution, then controls the motion of the stationary state and the remainder in new spaces, and devises new techniques for the multi-scale linearized analysis, which eventually enables the stability and modulation analysis around an approximate solution.

math.AP↗
arXiv2026

Classification of the dynamics of radial solutions to the 2D parabolic-elliptic Keller-Segel System

This note gives a complete classification of the asymptotic behavior of radial solutions to the two-dimensional parabolic-elliptic Keller-Segel system on the whole space, for general initial data in the large. We review previous separate results, and unify them within a single classification framework. Depending on the mass, the flow exhibits three distinct asymptotic regimes. For a subcritical mass, solutions converge toward the unique self-similar expander of same mass. At the critical mass $8\pi$, solutions concentrate in infinite time around the stationary state with a universal logarithmic rate. The determination of this behaviour was the last missing step for achieving a complete radial classification, and we prove it in a companion paper (in fact without radial assumption). For a supercritical mass, solutions undergo type II finite-time blow-up with an explicit universal asymptotic rate and the stationary state as profile. This trichotomy holds for all radial initial data with finite second momentum. For non-radial data or infinite second momentum, it is known that other dynamics can be possible; for each of these three universal regimes we review the known results showing how they persist.

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arXiv2025

Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system

We construct axially symmetric finite-time blow-up solutions to the three-dimensional Keller-Segel system. By adapting gluing techniques, we derive a precise asymptotic expansion for Type II singularities that generalizes the recent work of Hou, Nguyen, and Song. In our construction the mass concentrates along multiple rings and we obtain a refined expansion for the blow-up rate.

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arXiv2023

Existence of finite time blow-up in Keller-Segel system

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =Δu - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-Δ_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2π} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation}$ We show that there exists $\varepsilon>0$ such that for any $m$ satisfying $8π<m\le 8π+\varepsilon$ and any $k$ given points $q_{1},...,q_{k}$ in $\mathbb{R}^{2}$ there is an initial data $u_0^*$ of $(\star)$ for which the solution $u(x,t)$ blows-up in finite time as $t\to T$ with the approximate profile $$u(x,t)=\sum_{j=1}^{k}\frac{1}{λ_{j}^{2}(t)}U\left(\frac{x-ξ_{j}(t)}{λ_{j}(t)}\right)(1+o(1)), U(y)=\frac{8}{(1+|y|^{2})^{2}},$$ with $λ_{j}(t) \approx 2e^{-\frac{γ+2}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln(T-t)|}{2}}} $ where $γ=0.57721...$ is the Euler-Mascheroni constant, $ξ_{j}(t)\to q_{j}\in \mathbb{R}^{2}$ and such that $\int_{\mathbb{R}^2}u(x,t)dx=km.$ This construction generalizes the existence result of the stable blow-up dynamics recently proved in \cite{CGMN1,CGMN2}.

math.AP↗
arXiv2023

A sub-Riemannian maximum modulus theorem

In this note we prove a sub-Riemannian maximum modulus theorem in a Carnot group. Using a nontrivial counterexample, we also show that such result is best possible, in the sense that in its statement one cannot replace the right-invariant horizontal gradient with the left-invariant one.

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arXiv2021

On the limiting behaviour of some nonlocal seminorms: a new phenomenon

In this note we study the behaviour as $s\to 0^+$ of some semigroup based Besov seminorms associated with a non-symmetric and hypoelliptic diffusion with a drift. Our results generalise a previous one of Maz'ya and Shaposhnikova for the classical fractional Sobolev spaces $W^{s,p}$, and they also underscore a new phenomenon caused by the presence of the drift.

math.AP↗
arXiv2019

A chain rule for a class of evolutive nonlocal hypoelliptic equations

We prove a chain rule of local type for a class of fractional hypoelliptic equations of Kolmogorov-Fokker-Planck type. We introduce a semigroup based notion of nonlocal \emph{carré du champ} which works successfully in situations in which the infinitesimal generator of the semigroup itself does not necessarily possess a gradient. Our results extend and sharpen the original 2004 chain rule due to A. Córdoba and D. Córdoba.

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