On Global Regularity for Systems of Nonlinear Wave Equations with the Null-condition

where the index α gets summed over α “ 0, 1, . . . , 3, and the indices J,K are summed over 1, . . . ,N. Equations of this type play a prominent role in the theory of nonlinear waves, due to the special cancellation structure inherent in them, namely the Q0-null-form. This causes equations of this type to be amenable to better global regularity results than generic quadratic source terms, as witnessed for example in recent progress on the Wave Maps problem. The history of (1.1) goes at least back to the pivotal works [1], [5] by Christodoulou and Klainerman, respectively, which showed that in the physical space-dimension n “ 3, the nullcondition implies that smooth initial data p~u, ~utq|t“0 that decay sufficiently fast and are suitably small result in global in time smooth solutions. Of course this encapsulates much more equations than those of the form (1.1), and in particular certain quasilinear ones. The methods in the cited papers differ markedly, with the second one relying on the commuting vector fields method, while the first one used the Penrose compactification method. Both Klainerman’s and Christodoulou’s method apparently require p|x|∇qiu to be small for i “ 0, 1, . . . , 7, and a natural if highly non-trivial question is to enquire about the optimal condition ensuring global existence of solutions as well as sharp point wise decay bounds, in terms of the decay and number of derivatives of the data at |x| “ 8. A recent result by Pusateri [6] furnishes much weaker conditions resulting in global existence and sharp decay bounds, and in particular imposes only |x|2-weights on sufficiently many derivatives. His result relies on the recently developed ’space-time resonances’ method, see e. g. [7].