Remarks on the Well-Posedness of the Euler Equations in the Triebel–Lizorkin Spaces

Pointwise Decay of solutions to the energy critical nonlinear Schrödinger equations

“…The above lemma is useful for linear estimate. Motivated by [4], we derive the following improvement which is useful for the multilinear estimates.…”

“…It suffices to show∞ 0 e −is∆ |u|4 uds is convergent in L 1 , which can be obtained by the previous argument. Moreover, we haveu(t) − e it∆ φ + L ∞ ∞ t e i(t−s)△ |u(s)| 4 u(s)ds L ∞ 2t t e i(t−s)△ |u(s)| 4 u(s)ds L ∞ + ∞ 2t e i(t−s)△ |u(s)| 4 u(s)ds L ∞ 2t t e i(t−s)△ |u(s)| 4 u(s) H 2 ds + ∞ 2t |t − s| −3/2 |u(s)| 4 u(s) L 1 ds 2t t u(s) 4 L ∞ u(s) H 2 ds + ∞ 2t |s| −3/2 u(s) 3 L ∞ u(s)2 L 2 ds t −5 .…”