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

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 $W^{1,\infty}$.