Quintic NLS on R3: Difference between revisions

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** 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]]
* 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 focussing case from Glassey's virial identity.


[[Category:Schrodinger]]
[[Category:Schrodinger]]
[[Category:Equations]]
[[Category:Equations]]

Revision as of 23:00, 15 September 2006

The theory of the quintic NLS on is as follows.

  • Scaling is .
  • LWP is known for CaWe1990.
    • For the time of existence depends on the profile of the data as well as the norm.
    • For we have ill-posedness, indeed the 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 in the defocusing case CoKeStTkTa-p
    • For radial data this is in Bo1999b, Bo1999.
    • Blowup can occur in the focussing case from Glassey's virial identity.