Difference between revisions of "Cubic NLW/NLKG on R3"

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* Scaling is <math>s_c = 1/2</math>.
 
* Scaling is <math>s_c = 1/2</math>.
 
* LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[Bibliography#LbSo1995|LbSo1995]]; earlier references exist)
 
* LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[Bibliography#LbSo1995|LbSo1995]]; earlier references exist)
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** One can improve the critical space <math>H^{1/2}</math> to a slightly weaker Besov space [Pl-p2].
 
** One can improve the critical space <math>H^{1/2}</math> to a slightly weaker Besov space [Pl-p2].
 
** For <math>s<1/2</math> one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
 
** For <math>s<1/2</math> one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
* GWP for <math>s>3/4</math> [[references:KnPoVe-p2 KnPoVe-p2]] for defocussing NLKG.(An alternate proof is in [[Bibliography#GalPl2003|GalPl2003]]).
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* GWP for <math>s>3/4</math> [[Bibliography#KnPoVe-p2 |KnPoVe-p2]] for defocussing NLKG.(An alternate proof is in [[Bibliography#GalPl2003|GalPl2003]]).
 
** For <math>s\geq1</math> this is clear from energy conservation (for both NLKG and NLW).
 
** For <math>s\geq1</math> this is clear from energy conservation (for both NLKG and NLW).
 
** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general cubic non-linearities (and for either NLKG or NLW).
 
** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general cubic non-linearities (and for either NLKG or NLW).

Revision as of 15:57, 31 July 2006

  • Scaling is .
  • LWP for by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
    • When the time of existence depends on the profile of the data and not just on the norm.
    • One can improve the critical space to a slightly weaker Besov space [Pl-p2].
    • For one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
  • GWP for KnPoVe-p2 for defocussing NLKG.(An alternate proof is in GalPl2003).
    • For this is clear from energy conservation (for both NLKG and NLW).
    • One also has GWP and scattering for data with small norm for general cubic non-linearities (and for either NLKG or NLW).
    • In the defocussing case one has scattering for large data BaeSgZz1990, see also [Hi-p3].
    • Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
    • In the focussing case there is blowup from large data by the ODE method.
  • For periodic defocussing NLKG there is a weak turbulence effect in for (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} Kuk1995b.In particular cannot be a symplectic phase space for .