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

TL;DR

This paper demonstrates strong ill-posedness in the $L^ $infty$ space for certain 2D Euler equation perturbations involving Riesz transforms, showing solutions can grow rapidly from small initial data.

Contribution

It introduces a nonlinear model that captures the leading order effects, providing a precise growth rate that contrasts with previous linear analyses.

Findings

Solutions can grow arbitrarily large in short time from small initial data.

The nonlinear effects reduce the growth rate compared to the linear problem.

The growth rate aligns with the global regularity of smooth solutions.

Abstract

We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2d Euler equations of the form: \[\partial_t \omega + u\cdot\nabla\omega = R(\omega),\] 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.