Quintic NLS on R2: Difference between revisions

Latest revision as of 06:27, 21 July 2007

The theory of the quintic NLS on $R^{2}$ is as follows.

Scaling is $s_{c}=1/2\,$ .
LWP is known for $s\geq 1/2\,$ $s\geq 1/2\,$ CaWe1990.
- For $s=1/2\,$ the time of existence depends on the profile of the data as well as the norm.
- For $s<s_{c}\,$ we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
GWP for $s\geq 1\,$ $s\geq 1\,$ by Hamiltonian conservation.
- This has been improved to $s>1-\varepsilon$ in CoKeStTkTa2003b. This result can of course be improved further.
- Scattering in the energy space Na1999c
- One also has GWP and scattering for small $H^{1/2}\,$ data for any quintic non-linearity.

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 * LWP is known for <math>s \ge 1/2\,</math> [[CaWe1990]].
 ** For <math>s=1/2\,</math> the time of existence depends on the profile of the data as well as the norm.
-** For <math>s<s_c\,</math> we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
+** For <math>s<s_c\,</math> we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the [[virial identity]] and scaling.
 * GWP for <math>s \ge 1\,</math> by Hamiltonian conservation.
-** This has been improved to <math>s > 1-e\,</math>- in [[CoKeStTkTa2003b]]. This result can of course be improved further.
+** This has been improved to <math>s > 1-\varepsilon</math> in [[CoKeStTkTa2003b]]. This result can of course be improved further.
 ** Scattering in the energy space [[Na1999c]]
 ** One also has [[GWP]] and [[scattering]] for small <math>H^{1/2}\,</math> data for any quintic non-linearity.