arXiv subjects

Zachary Radke

Publications and source records attributed to Zachary Radke.

4 recordsLinked to original sources

arXiv2025

Ill-Posedness of the 2D Euler Equations in a Logarithmically Refined Critical Sobolev Space

In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D Euler equations for initial velocity in the critical Sobolev space $H^2(\mathbb{R}^2)$. In this work, we extend those results by demonstrating strong ill-posedness in logarithmically regularized spaces which are strictly contained in $H^2(\mathbb{R}^2)$ and which contain $H^s(\mathbb{R}^2)$ for all $s>2$. These spaces are constructed via application of a fractional logarithmic derivative to the critical Sobolev norm. We show that if the power $\alpha$ of the logarithmic derivative satisfies $\alpha\leq 1/2$, then the 2D Euler equations are strongly ill-posed.

math.AP↗

arXiv2025

Well-Posedness for the Euler Equations in Function Spaces of Generalized Smoothness

We consider the question of well-posedness for the incompressible Euler equations in generalized function spaces of the type $B^{s,\psi}_{p,q}(\mathbb{R}^d)$ and $F^{s,\psi}_{p,q}(\mathbb{R}^d)$ where $\psi$ is a slowly varying function in the Karamata sense and $s=d/p+1$. We prove that if $\psi$ grows fast enough, then there is a local in time solution to the Euler equations. We also establish a BKM-type criterion that allows us to obtain global existence in two dimensions.

math.AP↗

arXiv2025

Existence of Non-Decaying Solutions to the Generalized Surface Quasi-Geostrophic Equations

We establish the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in H\"older-Zygmund spaces $C^r(\mathbb{R}^2)$ for $r>1$ and uniformly local Sobolev spaces $H_{ul}^s(\mathbb{R}^2)$ for $s>2$.

math.AP↗

arXiv2024

Global Existence For A Nonlocal Multi-Species Aggregation-Diffusion Equation

We consider the question of global existence of smooth solutions to a multi-species aggregation-diffusion equation for a class of singular interaction kernels. We establish a smallness condition on the initial data which yields global existence of smooth solutions. We also give conditions on the species interaction which ensure that pointwise inequalities comparing species densities are preserved by the evolution.

math.AP↗