Download Communications in Mathematical Physics - Volume 256 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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The sets L(E,S),η are a cover of s(K) by finitely many disjoint compact subsets of X. For fixed (E, S) ∈ Qm , we now claim that there are compact graphs in G(E,S) , say M(E,S),1 , M(E,S),2 , . . C. Phillips and the sets r(M(E,S),1 ), r(M(E,S),2 ), . . , r(M(E,S),n ) are pairwise disjoint. 6) and their union is contained in G0 , we can take then the required graphs in G0 to be Mk = M(E,S),k . 3, card(S \ γ −1 S) ≤ card(S γ −1 S) ≤ c(Qm )ρ · card(S) < card(S) . n · card(T ) Therefore, with (S \ γ −1 S), R(E,S) = γ ∈T we have card(R(E,S) ) < n1 card(S).

3) |α|≤|β|≤2 allows solutions with smooth arbitrarily small initial data which blow up in finite time4 . The key to global existence for such equations was the null condition found by Klainerman, [K2]. The small data global existence result for the equations satisfying the null condition was established in [C1, K2]. 5 It can be shown however, that the Einstein vacuum equations in wave coordinates do not satisfy the null condition. Moreover, Choquet-Bruhat [CB3] showed that even without imposing a specific gauge the Einstein equations violate the null condition.

The generalized energy estimates are used with Minkowski vector fields {∂α , αβ = xα ∂β − xβ ∂α , S = x α ∂α }. For the equations satisfying the standard null condition uniform in time bounds on the generalized energies, combined with global Sobolev (Klainerman-Sobolev) inequalities, are sufficient to infer small data global existence. In our case however the generalized energies slowly grow in time (at the rate of t ε ) and need to be complemented by independent, not following from the global Sobolev inequalities, decay estimates.

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