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 work demonstrates that the Riesz transform is unbounded on $L^{\infty}$ via a new, elementary construction, and uses this to prove mild ill-posedness in $W^{1,\infty}$ for the 3D rotating Euler equations and for 2D Euler with partial damping. By coupling frequency-localized initial data with Lagrangian flow analysis, the authors show that arbitrarily small $W^{1,\infty}$-norm perturbations can yield large gradient growth in vanishingly short times, implying a discontinuity of the solution map in these critical spaces. The results extend beyond Euler-type flows to instability statements for the SQG and IPM models, highlighting a broad mechanism for non-uniqueness/non-continuous dependence in $L^{\infty}$-based frameworks. Overall, the paper advances the understanding of ill-posedness in geophysical and fluid models and identifies a unifying role for Riesz-type nonlocal operators in critical regularity settings.

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 $W^{1,\infty}$.

Theorems & Definitions (31)

• Definition 1.1
• Proposition 1.1
• Proposition 1.2
• Corollary 1.1: Linear ill-posedness
• Theorem 1.1
• Corollary 1.2
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Proposition 2.1: L-P decomposition, B