Download Near soliton evolution for equivariant Schrodinger maps in by Ioan Bejenaru, Daniel Tataru PDF

By Ioan Bejenaru, Daniel Tataru

The authors think of the Schrodinger Map equation in 2 1 dimensions, with values into S�. This admits a lowest strength regular country Q , specifically the stereographic projection, which extends to a dimensional family members of regular states via scaling and rotation. The authors end up that Q is volatile within the power house ?'. although, within the means of proving this in addition they express that in the equivariant type Q is good in a much better topology XC?'

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Extra info for Near soliton evolution for equivariant Schrodinger maps in two spatial dimensions

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In various contexts one removes this ambiguity by assigning a boundary condition either at ±∞ or at the initial time. In our case this issue does not arise since the constant is always fixed by the initial condition. We are primarily interested in the case p = 2. 6) 2 2 2 2 Q

H+ ) c) Assume that |j − h| 1 and |h − k| 1. 53) Pj Pkλ Ph = Pj Pk Ph + cjkh cjkh = k+ − 2 l−N g l (λ)Qljkh , 2 k with g l uniformly bounded in C N and Qljkh L2 →L2 ≤ 1. ˆ jkh of Proof. 8. NONRESONANT QUADRILINEAR FORMS 41 with χlj , χlh smooth dyadic bump functions, and g smooth. 11. b) The proof is similar to the one in case (a), with the only difference that we need to consider the contribution of the diagonal term in exactly one of the kernels K1λ (ξ, η) and Kλ1 (η, ζ). 46) in order to claim that if s is smooth in η then b(λ1 , λ2 , ξ, η) s(η)dη ξ 2 − η2 is smooth in ξ.

5. 25) Proof. 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.

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