Quintic NLW/NLKG on R2: Difference between revisions
From DispersiveWiki
Jump to navigationJump to search
No edit summary |
No edit summary |
||
| Line 2: | Line 2: | ||
* LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[LbSo1995]]; earlier references exist) | * 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. | ||
** For <math>s<s_c</math> one has instantaneous blowup in the focusing case, and unbounded growth of <math>H^s</math> norms in the defocusing case [[CtCoTa-p2]] | ** For <math>s<s_c</math> one has instantaneous blowup in the focusing case, and unbounded growth of <math>H^s</math> norms in the defocusing case ([[CtCoTa-p2]]) | ||
* GWP for <math>s > 3/4</math> for defocussing NLW/NLKG [[Fo-p]] | * GWP for <math>s > 3/4</math> for defocussing NLW/NLKG ([[Fo-p]]) | ||
** For <math>s \geq 1</math> this follows energy conservation. | ** For <math>s \geq 1</math> this follows energy conservation. | ||
** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general quintic non-linearities (and for either NLW or NLKG). | ** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general quintic non-linearities (and for either NLW or NLKG). | ||
Latest revision as of 07:10, 2 August 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.
- For one has instantaneous blowup in the focusing case, and unbounded growth of norms in the defocusing case (CtCoTa-p2)
- GWP for for defocussing NLW/NLKG (Fo-p)
- For this follows energy conservation.
- One also has GWP and scattering for data with small norm for general quintic non-linearities (and for either NLW or NLKG).
- In the focussing case there is blowup from large data by the ODE method.
