arXiv subjects

Xuan-Truong Vu

Publications and source records attributed to Xuan-Truong Vu.

4 recordsLinked to original sources

arXiv2026

Regularity of solution maps of the generalized surface quasi-geostrophic equations

We study regularity properties of the data-to-solution maps of the family of generalized surface quasi-geostrophic equations which includes both the 2D incompressible Euler and the standard surface quasi-geostrophic equations. We prove that the Lagrangian solution maps, interpreted as Riemannian exponential maps on the group of exact Sobolev class diffeomorphisms, are real analytic and, consequently, the Cauchy problems are locally well-posed in the sense of Hadamard. On the other hand, we also show that the corresponding Eulerian solution maps are nowhere locally uniformly continuous on bounded subsets in the Sobolev topology and fail to be continuous in the standard (large-) Hölder topologies. These results sharpen earlier theorems and further highlight the striking dichotomy between regularity properties of the solution maps in the Lagrangian and Eulerian formulations.

math.AP↗

arXiv2025

Instability of Data-to-Solution Map for the Log-Regularized 2D Euler System

In this paper, we study the logarithmically regularized $2$D Euler system \eqref{e1}, which is derived by regularizing the Euler equation for the vorticity. We establish local well-posedness of the logarithmically regularized $2$D Euler equations in the subcritical space $H^s(\mathbb{R}^2)$ with $s>2$ for $γ\ge 0$. Furthermore, we show that for $γ$ close to $0$, the data-to-solution map is not uniformly continuous in the Sobolev $H^s(\mathbb{R}^2)$ topology for any $s>2$.

math.AP↗

arXiv2023

Almost sure behavior of the zeros of iterated derivatives of random polynomials

Let $Z_1,\, Z_2,\dots$ be independent and identically distributed complex random variables with common distribution $μ$ and set $$ P_n(z) := (z - Z_1)\cdots (z - Z_n)\,. $$ Recently, Angst, Malicet and Poly proved that the critical points of $P_n$ converge in an almost-sure sense to the measure $μ$ as $n$ tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed $k\in \mathbb{N}$, the empirical measure of zeros of the $k$th derivative of $P_n$ converges to $μ$ in the almost sure sense, as conjectured by Angst-Malicet-Poly.

math.PR↗

arXiv2022

Zeros of a growing number of derivatives of random polynomials with independent roots

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $μ$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $μ$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy who proved the fixed $k$ case.

math.PR↗