Strong ill-posedness in L∞ for the Riesz transform problem

Elgindi, Tarek M.; Shikh Khalil, Karim R.

doi:10.2140/apde.2025.18.715

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Strong ill-posedness in $L^\infty$ for the Riesz transform problem

Tarek M. Elgindi and Karim R. Shikh Khalil

Vol. 18 (2025), No. 3, 715–741

DOI: 10.2140/apde.2025.18.715

Abstract

We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2-dimensional Euler equations of the form

\partial_{t} ω + u \cdot \nabla ω = R (ω),

where $R$ is any nontrivial 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. 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.

Keywords

Euler equations, ill-posedness, Euler–Riesz equations

Mathematical Subject Classification

Primary: 35Q31, 35Q35

Milestones

Received: 4 October 2022

Revised: 1 November 2023

Accepted: 21 December 2023

Published: 3 March 2025

Authors

Tarek M. Elgindi
Department of Mathematics Duke University Durham, NC United States

Karim R. Shikh Khalil
Department of Mathematics Duke University Durham, NC United States

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Vol. 18, No. 3, 2025