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An evidence of temporal dis-continuity of the solution in $F^s_{1,\infty}(\mathbb{R}^d)$ is presented, which implies the ill-posedness of theCauchy problem for the Euler equations. Continuity and weak-type continuity ofthe solutions in related spaces are also discussed.

Temporal regularity of the solution to the incompressible Euler  equations in the end-point critical Triebel-Lizorkin space $F^{d+1}_{1,  \infty}(\mathbb{R}^d)$