Strong Ill-Posedness in $L^\infty$ for the Riesz Transform Problem

TL;DR

The paper addresses strong ill-posedness in $L^{\infty}$ for the 2D Euler equations with a second-order Riesz forcing $R$, studying $\partial_t \omega+u\cdot\nabla\omega=R(\omega)$. It derives a leading-order nonlinear model via a Biot-Savart decomposition, obtaining a nonlinear transport equation with projections $L_s$ and $L_c$, and proves sharp $L^{\infty}$ growth bounds of the form $|\omega(t)|_{L^{\infty}} \ge |\omega_0|_{L^{\infty}}+c\log\bigl(1+(c/\alpha)t\bigr)$ on a short time scale $T(\alpha)=c\alpha|\log\alpha|$, while controlling the remainder to link the full dynamics to the leading-order behavior. The remainder analysis shows the full solution remains close to the leading-order dynamics on $[0,T(\alpha)]$, establishing strong ill-posedness yet consistent with potential global regularity. The results extend to any second-order Riesz transform via simple coordinate changes, clarifying how nonlinear transport interacts with nonlocal forcing and providing a rigorous mechanism for finite-time $L^{\infty}$ growth in this setting.

Abstract

We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2d Euler equations of the form: \[\partial_t ω+ u\cdot\nablaω= R(ω),\] where $R$ is any non-trivial second order Riesz transform. Namely, we prove that there exist smooth solutions that are initially small in $L^{\infty}$ but become arbitrarily large in short time. Previous works in this direction relied on the strong ill-posedness of the linear problem, viewing the transport term perturbatively, which only led to mild growth. In this work we derive a nonlinear model taking all of the leading order effects into account to determine the precise pointwise growth of solutions for short time. Interestingly, the Euler transport term does counteract the linear growth so that the full nonlinear equation grows an order of magnitude less than the linear one. In particular, the (sharp) growth rate we establish is consistent with the global regularity of smooth solutions.

Theorems & Definitions (30)

• Theorem 1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof