A new proof of unboundedness of Riesz operator in $L^\infty$ and applications to mild ill-posedness in $W^{1,\infty}$ of the Euler type equations

TL;DR

This paper provides a new proof of the unboundedness of the Riesz operator in $L^ olinebreak^ olinebreak ext{infty}$ and explores the mild ill-posedness of Euler equations and related models in $W^{1, olinebreak ext{infty}}$, revealing instability phenomena.

Contribution

It introduces a novel, simplified proof of Riesz operator unboundedness and demonstrates ill-posedness of Euler equations in $W^{1, olinebreak ext{infty}}$ without vorticity formulation, a first in this context.

Findings

Unboundedness of Riesz operator in $L^ olinebreak^ olinebreak ext{infty}$ established

Mild ill-posedness of 3D rotating Euler equations in $W^{1, olinebreak ext{infty}}$

Instability of perturbations in 2D surface quasi-geostrophic and porous medium systems

Abstract

In this paper, we first present a new and simple proof of unboundedness of Riesz operator in $L^\infty$ and then establish the mild ill-posedness in $W^{1,\infty}$ of 3D rotating Euler equations and 2D Euler equations with partial damping. To the best of our knowledge, our work is the first one addressing the ill-posedness issue on the rotating Euler equations in $W^{1,\infty}$ without the vorticity formulation. As a further application, we prove the instability of perturbations for the 2D surface quasi-geostrophic equation and porous medium system in $W^{1,\infty}$.