The Coriolis effect-induced $B_{\infty ,r}^s$ and $B_{1,r}^s$ ill-posedness in the 3D rotating Euler equations

Abstract

In this paper, we establish the ill-posedness of the 3D rotating Euler equations in endpoint $L^{\infty }$ and $L^1$-based spaces. More precisely, our aim is two fold: (1) we prove an endpoint ill-posedness in any $B_{\infty ,r}^s$ of the velocity formulation for the 3D rotating Euler equations. To the best of our knowledge, our work is the first one addressing the ill-posedness issue on the rotating Euler equations in any $L^{\infty }$-based framework without the vorticity formulation. These are in stark contrast to existing well-posedness results for the classical Euler equations in endpoint Besov spaces $B_{\infty ,1}^1({\mathbb{R}}^3)$ and thus we identify a structural distinction between the 3D rotating and classical Euler equations. (2) we prove an endpoint ill-posedness in any $B_{1,r}^s$ of the vorticity formulation for the 3D rotating Euler equations, which seems to be the first ill-posedness result in $L^1$-type spaces although the currently-known results on the well-posedness were given in the Besov space $B_{1,1}^{d+1}({\mathbb{R}}^d)$ for the velocity equation and in the Sobolev space $W^{d,1}({\mathbb{R}}^d)$ for the vorticity equation. The Coriolis effect-induced $B_{\infty ,r}^s$ and $B_{1,r}^s$ ill-posedness phenomenon for the 3D rotating Euler equations, which may be the first work that precisely identifies a rigorous distinction in the theory of well-posedness on the 3D rotating and classical Euler equations.

arXiv commentCorrected some typos