By Ioan Bejenaru, Daniel Tataru
The authors think about the Schrodinger Map equation in 2 1 dimensions, with values into Sï¿½. This admits a lowest strength regular country Q , particularly the stereographic projection, which extends to a dimensional family members of regular states via scaling and rotation. The authors turn out that Q is volatile within the strength area ?'. although, within the means of proving this additionally they convey that in the equivariant classification Q is strong in a far better topology XC?'
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Extra resources for Near soliton evolution for equivariant Schrodinger maps in two spatial dimensions
57 58 6. 1. The spaces WS [λ], WN [λ] Here we deﬁne the [λ] type spaces as counterparts of the spaces from the previous section which take into account time-variable Littlewood-Paley projectors. We begin as usual with a dyadic decomposition, but with respect to the λ dependent frame, Pkλ ψ. 1). For LX initial data, on the other hand, we need to replace the l2 dyadic summation with the same summation as in the LX norm. Hence we deﬁne ⎞ 12 ⎛ 1 P λ ψ l2 X + ⎝ ψ W X [λ] = Pkλ ψ 2l2 X ⎠ 2k |k| k k<0 k≥0 All of the above spaces l2 X [λ] and W X [λ] have their ﬁnite time interval counterpart l2 X [λ](I) and W X [λ](I), which are obtained by using l2 X (I) instead of l2 X in the above deﬁnitions.
This corresponds to ﬁelds ψ which satisfy ψ LX 1. 3. 2. 18) 1 2 r f L∞ f X f X L2 f L4 X Proof. 3. Assume fk ∈ L2 is localized at H-frequency 2k . 21) 2 2 fk ∂r fk (r) 2−k L2 , 2k fk L2 (A≥−k ) fk L2 , r 2−k r L2 . Proof. 5) and the Cauchy-Schwarz inequality. 20) follows from the bound |φξ | r − 2 for r 2−k . 22) Lfk 2k fk L2 L2 We now prove the embedding X ⊂ H˙ e1 . Due to the straightforward bound f X and the ODE estimate Lf L2 ∂r f L2 + r −1 f |f (1)| + Lf L2 L2 , it suﬃces to show that |f (1)| f X .
The second embedding is trivial. 26) LX f L1 1 It suﬃces to consider the case when f is a Dirac mass, f = δr=R . 26) follows. 3. 6. 28) h1 Lf f LX LX g X + h3 Lf X f LX X Proof. 23) is it enough to prove that if f, g ∈ X then f · g ∈ L2 and L(f · g) ∈ L2 . 16). 29) L2 r −1 f L2 g f L∞ g ˙1 H e ˙1 H e and the proof of the algebra property is complete. 27). 25). 30). We are then left with estimating the low frequency contributions P≤0 f LP≤0 g. e. when f is replaced by fk = Pk f and g by gj = Pj g with k, j < 0.
Near soliton evolution for equivariant Schrodinger maps in two spatial dimensions by Ioan Bejenaru, Daniel Tataru