Quintic NLS on R2

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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.