Two-speed DDNLW: Difference between revisions

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==Two-speed DDNLW==
One can consider [[two-speed wave equations|two-speed variants]] of [[DDNLW]]


* One can consider two-speed variants of DDNLW ([#two-speed see Overview]), when both F and G have the form  G (U) DU DU.
<center><math>\Box u = F(U) DU DU, ~\Box_s v = G(U) DU DU</math></center>
* The Strichartz and energy estimates carry over without difficulty to this setting. The results obtained by X^{s,\theta} 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 DDNLS arises in elasticity (more on this to be added in later).
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.
----   [[Category:Equations]]
* 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.