Cubic NLW/NLKG on R3: Difference between revisions
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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. [[ | * LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[LbSo1995]]; earlier references exist) | ||
** When <math>s=1/2</math> the time of existence depends on the profile of the data and not just on the norm. | ** When <math>s=1/2</math> the time of existence depends on the profile of the data and not just on the norm. | ||
** 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> [[ | * GWP for <math>s>3/4</math> ([[KnPoVe-p2]]) for defocussing NLKG.(An alternate proof is in [[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). | ||
** In the defocussing case one has scattering for large <math>H^1</math> data [[ | ** In the defocussing case one has scattering for large <math>H^1</math> data ([[BaeSgZz1990]]), see also ([[Hi-p3]]). | ||
** Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW. | ** 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. | ** 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 <math>H^s</math> for <math>s > 5</math> (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} [[ | * For periodic defocussing NLKG there is a weak turbulence effect in <math>H^s</math> for <math>s > 5</math> (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} [[Kuk1995b]].In particular <math>H^s</math> cannot be a symplectic phase space for <math>s > 5</math>. | ||
[[Category:Wave]] | |||
[[Category:Equations]] | |||
Latest revision as of 04:55, 2 August 2006
- Scaling is .
- LWP for by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
- 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 .
