Two-speed DDNLW: Difference between revisions
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One can consider [[two-speed wave equations|two-speed variants]] of [[DDNLW]] | |||
<center><math>\Box u = F(U) DU DU, ~\Box_s v = G(U) DU DU</math></center> | |||
where <math>U = (u,v)</math> and <math>F,G</math> are tensor-valued nonlinearities. | |||
* The Strichartz and energy estimates carry over without difficulty to this setting. The results obtained by [[X^s,b spaces|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]]. | |||
[[Category:wave]] | |||
[[Category:Equations]] | |||
Latest revision as of 21:13, 30 July 2006
One can consider two-speed variants of DDNLW
where and 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.
