Read e-book online Communications in Mathematical Physics - Volume 211 PDF

By A. Jaffe (Chief Editor)

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22) 54 A. Constantin, L. Molinet and define Gn (t) := 3 R (ρn ∗ ux )(ρn ∗ u2 ) dx − 2 + R R [ρn ∗ ux ] [ρn,x ∗ (uux )] dx (ρn ∗ ux ) (ρn ∗ u2x ) dx, t ∈ R+ , n ≥ 1. e. 23) Gn (t) −→ 0. 23) that En (t) − En (0) = t 0 Gn (s) ds, t ∈ R+ , n ≥ 1. 19). 26) as u, ux ∈ L∞ (R+ × R). 26) force lim [En (t) − En (0)] = 0, t ∈ R+ , n→∞ on account of Lebesgue’s dominated convergence theorem. 6). This proves that u ∈ C(R+ ; H 1 (R)) and E is conserved along our solution. 3) and Young’s inequality, it is now obvious that u ∈ C 1 (R+ ; L2 (R)).

Note that this notation agrees with our tensor notation above. Also note that for 34 F. Xu any covariant representation πx of A(I ), we can define a localized sector x ⊗ 1 of N (I ) in the same way as in the case when πx = πi,α . Each covariant representation π i of LG gives rise to an endomorphism σi ∈ End(N(I )) and (cf. Subsect. 1) ρσi ρ¯ = γi = (i, α) ⊗ (α), α where the summation is over those α such that (i, α) ∈ exp. So by the properties of statistical dimensions (cf. 1) di dρ2 = α d(i,α) dα .

Step iv) As u ∈ Cw (R+ , H 1 (R)), to conclude that u ∈ C(R+ , H 1 (R)) it is enough to show that the functional E(u(t)) = u(t, ·) 2H 1 (R) is conserved in time. Indeed, if this holds, then u(t) − u(s) 2 H 1 (R) = u(t) 2 H 1 (R) − 2 u(t), u(s) = 2 u0 2 H 1 (R) − 2 u(t), u(s) H 1 (R) + u(s) 2 H 1 (R) , t, s ∈ R+ , H 1 (R) and the scalar product in the last line converges to u(t) 2H 1 (R) = u0 2H 1 (R) as s → t. The conservation of E(u) in time is proved by a regularization technique. e. t ∈ R+ , 1 2 u ) = 0, n ≥ 1.

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Communications in Mathematical Physics - Volume 211 by A. Jaffe (Chief Editor)


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