Quintic NLS on R3: Difference between revisions

Latest revision as of 06:28, 21 July 2007

The theory of the quintic NLS on $\mathbb {R} ^{3}$ is as follows.

Scaling is $s_{c}=1\,$ .
LWP is known for $s\geq 1\,$ $s\geq 1\,$ CaWe1990.
- For $s=1\,$ 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 and scattering for $s\geq 1\,$ $s\geq 1\,$ in the defocusing case CoKeStTkTa-p
- For radial data this is in Bo1999b, Bo1999.
- Blowup can occur in the focusing case from Glassey's virial identity.

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 * LWP is known for <math>s \ge 1\,</math> [[CaWe1990]].
 ** For <math>s=1\,</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 <math>H^s\,</math> 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 <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the [[focusing]] case we have instantaneous blowup from the [[virial identity]] and scaling.
 * GWP and scattering for <math>s\ge 1\,</math> in the defocusing case [[CoKeStTkTa-p]]
-** For radial data this is in [[Bo-p]], [[Bo1999]].
+** For radial data this is in [[Bo1999b]], [[Bo1999]].
-** Blowup can occur in the focussing case from Glassey's virial identity.
+** Blowup can occur in the focusing case from Glassey's virial identity.
 [[Category:Schrodinger]]
 [[Category:Equations]]