Zachary Radke

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math.AP2025

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

Elaine Cozzi, Nicholas Harrison, Zachary Radke

In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D Euler equations for initial velocity in the critical Sobolev space $H^{2} (R^{2})$ . In this work…

math.AP2025

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

Nicholas Harrison, Zachary Radke

We consider the question of well-posedness for the incompressible Euler equations in generalized function spaces of the type $B_{p, q}^{s, ψ} (R^{d})$ and $F^{s,ψ}_{p,q}(\mathb…

math.AP2025

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

Zachary Radke

We establish the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in Hölder-Zygmund spaces $C^{r} (R^{2})$ f…

math.AP2024

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

Elaine Cozzi, Zachary Radke

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 small…