Two-speed DDNLW

One can consider two-speed variants of DDNLW

$\Box u=F(U)DUDU,~\Box _{s}v=G(U)DUDU$

where $U=(u,v)$ and $F,G$ are tensor-valued nonlinearities.

• The Strichartz and energy estimates carry over without difficulty to this setting. The results obtained by X^{s,b} estimates change, however. The null forms are no longer as useful, however the estimates are usually more favourable because of the transversality of the two light cones. Of course, if F contains DuDu or G contains DvDv then one cannot do any better than the one-speed case.
• For d=2 one can obtain LWP for the near-optimal range s>3/2 when F does not contain DuDu and G does not contain DvDv Tg-p.
• For d=1 one can obtain LWP for the near-optimal range s>1 when F does not contain DuDu and G does not contain DvDv Tg-p.
• For d=3 one can obtain GWP for small compactly supported data for quasilinear equations with multiple speeds, as long as the nonlinearity has no explicit dependence on U KeSmhSo-p3

A special case of two-speed DDNLW arises in elasticity.