Quintic NLS on R2: Difference between revisions

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* LWP is known for <math>s \ge 1/2\,</math> [[CaWe1990]].
* 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=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.
* 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]]
** 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.
** One also has [[GWP]] and [[scattering]] for small <math>H^{1/2}\,</math> data for any quintic non-linearity.

Latest revision as of 06:27, 21 July 2007

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 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 by Hamiltonian conservation.
    • This has been improved to in CoKeStTkTa2003b. This result can of course be improved further.
    • Scattering in the energy space Na1999c
    • One also has GWP and scattering for small data for any quintic non-linearity.