Energy Solution to the Chern-Simons-Schrödinger Equations

and Applied Analysis 3 The following type of Strichartz estimate was used in [19, 20] for the study of the Benjamin-Ono equation. We refer to [12] for the counterpart to the Schrödinger equation. Lemma 5. Let T ≤ 1 and V be a solution to the equation i∂ t V + ΔV = F 1 + F 2 , (t, x) ∈ (0, T) ×R 2 . (15) Then, for δ ∈ R and ε > 0, one has 󵄩󵄩󵄩󵄩 J δ V 󵄩󵄩󵄩󵄩L p T L q ≲ ‖V‖ L ∞ T H δ+1/2+ε + 󵄩󵄩󵄩󵄩F1 󵄩󵄩󵄩󵄩L2 T H δ−1/2 + 󵄩󵄩󵄩󵄩F2 󵄩󵄩󵄩󵄩L1 T H δ , (16) where 1/p + 1/q = 1/2 and 2 ≤ q < ∞. We use the following Gagliardo-Nirenberg inequality with the specific constant [21], especially for the proof of Theorem 2. Lemma 6. For 2 ≤ q < ∞, one has ‖u‖Lq(R2) ≤ (4π) (2−q)/2q ( q